# Finite word¶

AUTHORS:

• Arnaud Bergeron
• Amy Glen
• Sébastien Labbé
• Franco Saliola
• Julien Leroy (March 2010): reduced_rauzy_graph

EXAMPLES:

## Creation of a finite word¶

Finite words from python strings, lists and tuples:

sage: Word("abbabaab")
word: abbabaab
sage: Word([0, 1, 1, 0, 1, 0, 0, 1])
word: 01101001
sage: Word( ('a', 0, 5, 7, 'b', 9, 8) )
word: a057b98


Finite words from functions:

sage: f = lambda n : n%3
sage: Word(f, length=13)
word: 0120120120120


Finite words from iterators:

sage: from itertools import count
sage: Word(count(), length=10)
word: 0123456789

sage: Word( iter('abbccdef') )
word: abbccdef


Finite words from words via concatenation:

sage: u = Word("abcccabba")
sage: v = Word([0, 4, 8, 8, 3])
sage: u * v
word: abcccabba04883
sage: v * u
word: 04883abcccabba
sage: u + v
word: abcccabba04883
sage: u^3 * v^(8/5)
word: abcccabbaabcccabbaabcccabba04883048


Finite words from infinite words:

sage: vv = v^Infinity
sage: vv[10000:10015]
word: 048830488304883


Finite words in a specific combinatorial class:

sage: W = Words("ab")
sage: W
Words over {'a', 'b'}
sage: W("abbabaab")
word: abbabaab
sage: W(["a","b","b","a","b","a","a","b"])
word: abbabaab
sage: W( iter('ababab') )
word: ababab


Finite word as the image under a morphism:

sage: m = WordMorphism({0:[4,4,5,0],5:[0,5,5],4:[4,0,0,0]})
sage: m(0)
word: 4450
sage: m(0, order=2)
word: 400040000554450
sage: m(0, order=3)
word: 4000445044504450400044504450445044500550...


## Functions and algorithms¶

There are more than 100 functions defined on a finite word. Here are some of them:

sage: w = Word('abaabbba'); w
word: abaabbba
sage: w.is_palindrome()
False
sage: w.is_lyndon()
False
sage: w.number_of_factors()
28
sage: w.critical_exponent()
3

sage: print w.lyndon_factorization()
(ab, aabbb, a)
sage: print w.crochemore_factorization()
(a, b, a, ab, bb, a)

sage: st = w.suffix_tree()
sage: st
Implicit Suffix Tree of the word: abaabbba
sage: st.show(word_labels=True)

sage: T = words.FibonacciWord('ab')
sage: T.longest_common_prefix(Word('abaabababbbbbb'))
word: abaababa


As matrix and many other sage objects, words have a parent:

sage: u = Word('xyxxyxyyy')
sage: u.parent()
Words

sage: v = Word('xyxxyxyyy', alphabet='xy')
sage: v.parent()
Words over {'x', 'y'}


## Factors and Rauzy Graphs¶

Enumeration of factors, the successive values returned by it.next() can appear in a different order depending on hardware. Therefore we mark the three first results of the test random. The important test is that the iteration stops properly on the fourth call:

sage: w = Word([4,5,6])^7
sage: it = w.factor_iterator(4)
sage: it.next() # random
word: 6456
sage: it.next() # random
word: 5645
sage: it.next() # random
word: 4564
sage: it.next()
Traceback (most recent call last):
...
StopIteration


The set of factors:

sage: sorted(w.factor_set(3))
[word: 456, word: 564, word: 645]
sage: sorted(w.factor_set(4))
[word: 4564, word: 5645, word: 6456]
sage: w.factor_set().cardinality()
61


Rauzy graphs:

sage: f = words.FibonacciWord()[:30]
sage: f.rauzy_graph(4)
Looped digraph on 5 vertices
sage: f.reduced_rauzy_graph(4)
Looped multi-digraph on 2 vertices


Left-special and bispecial factors:

sage: f.number_of_left_special_factors(7)
1
sage: f.bispecial_factors()
[word: , word: 0, word: 010, word: 010010, word: 01001010010]

class sage.combinat.words.finite_word.CallableFromListOfWords

Bases: tuple

A class to create a callable from a list of words. The concatenation of a list of words is obtained by creating a word from this callable.

class sage.combinat.words.finite_word.Factorization

Bases: list

A list subclass having a nicer representation for factorization of words.

TESTS:

sage: f = sage.combinat.words.finite_word.Factorization()
True

class sage.combinat.words.finite_word.FiniteWord_class

x.__init__(...) initializes x; see help(type(x)) for signature

BWT()

Returns the Burrows-Wheeler Transform (BWT) of self.

The Burrows-Wheeler transform of a finite word $$w$$ is obtained from $$w$$ by first listing the conjugates of $$w$$ in lexicographic order and then concatenating the final letters of the conjugates in this order. See [1].

EXAMPLES:

sage: Word('abaccaaba').BWT()
word: cbaabaaca
sage: Word('abaab').BWT()
word: bbaaa
sage: Word('bbabbaca').BWT()
word: cbbbbaaa
sage: Word('aabaab').BWT()
word: bbaaaa
sage: Word().BWT()
word:
sage: Word('a').BWT()
word: a


REFERENCES:

apply_permutation_to_letters(permutation)

Return the word obtained by applying permutation to the letters of the alphabet of self.

EXAMPLES:

sage: w = Words('abcd')('abcd')
sage: p = [2,1,4,3]
sage: w.apply_permutation_to_letters(p)
sage: u = Words('dabc')('abcd')
sage: u.apply_permutation_to_letters(p)
word: dcba
sage: w.apply_permutation_to_letters(Permutation(p))
sage: w.apply_permutation_to_letters(PermutationGroupElement(p))

apply_permutation_to_positions(permutation)

Return the word obtained by permuting the positions of the letters in self.

EXAMPLES:

sage: w = Words('abcd')('abcd')
sage: w.apply_permutation_to_positions([2,1,4,3])
sage: u = Words('dabc')('abcd')
sage: u.apply_permutation_to_positions([2,1,4,3])
sage: w.apply_permutation_to_positions(Permutation([2,1,4,3]))
sage: w.apply_permutation_to_positions(PermutationGroupElement([2,1,4,3]))
sage: Word([1,2,3,4]).apply_permutation_to_positions([3,4,2,1])
word: 3421

balance()

Returns the balance of self.

The balance of a word is the smallest number $$q$$ such that self is $$q$$-balanced [1].

A finite or infinite word $$w$$ is said to be $$q$$-balanced if for any two factors $$u$$, $$v$$ of $$w$$ of the same length, the difference between the number of $$x$$‘s in each of $$u$$ and $$v$$ is at most $$q$$ for all letters $$x$$ in the alphabet of $$w$$. A $$1$$-balanced word is simply said to be balanced. See Chapter 2 of [2].

OUTPUT:

integer

EXAMPLES:

sage: Word('1111111').balance()
0
sage: Word('001010101011').balance()
2
sage: Word('0101010101').balance()
1

sage: w = Word('11112222')
sage: w.is_balanced(2)
False
sage: w.is_balanced(3)
False
sage: w.is_balanced(4)
True
sage: w.is_balanced(5)
True
sage: w.balance()
4


TESTS:

sage: Word('1111122222').balance()
5
sage: Word('').balance()
0
sage: Word('1').balance()
0
sage: Word('12').balance()
1
sage: Word('1112').balance()
1


REFERENCES:

• [1] I. Fagnot, L. Vuillon, Generalized balances in Sturmian words, Discrete Applied Mathematics 121 (2002), 83–101.
• [2] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
bispecial_factors(n=None)

Returns the bispecial factors (of length n).

A factor $$u$$ of a word $$w$$ is bispecial if it is right special and left special.

INPUT:

• n - integer (optional, default: None). If None, it returns all bispecial factors.

OUTPUT:

A list of words.

EXAMPLES:

sage: w = words.FibonacciWord()[:30]
sage: w.bispecial_factors()
[word: , word: 0, word: 010, word: 010010, word: 01001010010]

sage: w = words.ThueMorseWord()[:30]
sage: for i in range(10): print i, sorted(w.bispecial_factors(i))
0 [word: ]
1 [word: 0, word: 1]
2 [word: 01, word: 10]
3 [word: 010, word: 101]
4 [word: 0110, word: 1001]
5 []
6 [word: 011001, word: 100110]
7 []
8 [word: 10010110]
9 []

bispecial_factors_iterator(n=None)

Returns an iterator over the bispecial factors (of length n).

A factor $$u$$ of a word $$w$$ is bispecial if it is right special and left special.

INPUT:

• n - integer (optional, default: None). If None, it returns an iterator over all bispecial factors.

EXAMPLES:

sage: w = words.ThueMorseWord()[:30]
sage: for i in range(10):
...     for u in sorted(w.bispecial_factors_iterator(i)):
...         print i,u
0
1 0
1 1
2 01
2 10
3 010
3 101
4 0110
4 1001
6 011001
6 100110
8 10010110

sage: key = lambda u : (len(u), u)
sage: for u in sorted(w.bispecial_factors_iterator(), key=key): u
word:
word: 0
word: 1
word: 01
word: 10
word: 010
word: 101
word: 0110
word: 1001
word: 011001
word: 100110
word: 10010110

border()

Returns the longest word that is both a proper prefix and a proper suffix of self.

EXAMPLES:

sage: Word('121212').border()
word: 1212
sage: Word('12321').border()
word: 1
sage: Word().border() is None
True

charge(check=True)

Returns the charge of self. This is defined as follows.

If $$w$$ is a permutation of length $$n$$, (in other words, the evaluation of $$w$$ is $$(1, 1, \dots, 1)$$), the statistic charge($$w$$) is given by $$\sum_{i=1}^n c_i(w)$$ where $$c_1(w) = 0$$ and $$c_i(w)$$ is defined recursively by setting $$p_i$$ equal to $$1$$ if $$i$$ appears to the right of $$i-1$$ in $$w$$ and $$0$$ otherwise. Then we set $$c_i(w) = c_{i-1}(w) + p_i$$.

EXAMPLES:

sage: Word([1, 2, 3]).charge()
3
sage: Word([3, 5, 1, 4, 2]).charge() == 0 + 1 + 1 + 2 + 2
True


If $$w$$ is not a permutation, but the evaluation of $$w$$ is a partition, the charge of $$w$$ is defined to be the sum of its charge subwords (each of which will be a permutation). The first charge subword is found by starting at the end of $$w$$ and moving left until the first $$1$$ is found. This is marked, and we continue to move to the left until the first $$2$$ is found, wrapping around from the beginning of the word back to the end, if necessary. We mark this $$2$$, and continue on until we have marked the largest letter in $$w$$. The marked letters, with relative order preserved, form the first charge subword of $$w$$. This subword is removed, and the next charge subword is found in the same manner from the remaining letters. In the following example, $$w1, w2, w3$$ are the charge subwords of $$w$$.

EXAMPLE:

sage: w = Word([5,2,3,4,4,1,1,1,2,2,3])
sage: w1 = Word([5, 2, 4, 1, 3])
sage: w2 = Word([3, 4, 1, 2])
sage: w3 = Word([1, 2])
sage: w.charge() == w1.charge() + w2.charge() + w3.charge()
True


Finally, if $$w$$ does not have partition content, we apply the Lascoux-Schutzenberger standardization operators $$s_i$$ in such a manner as to obtain a word with partition content. (The word we obtain is independent of the choice of operators.) The charge is then defined to be the charge of this word:

sage: Word([3,3,2,1,1]).charge()
0
sage: Word([1,2,3,1,2]).charge()
2


Note that this differs from the definition of charge given in Macdonald’s book. The difference amounts to a choice of reading a word from left-to-right or right-to-left. The choice in Sage was made to agree with the definition of a reading word of a tableau in Sage, and seems to be the more common convention in the literature.

REFERENCES:

[1] Ian Macdonald, Symmetric Functions and Hall Polynomials second edition, 1995, Oxford University Press

[2] A. Lascoux, L. Lapointe, and J. Morse. Tableau atoms and a new Macdonald positivity conjecture. Duke Math Journal, 116 (1), 2003. Available at: [http://arxiv.org/abs/math/0008073]

[3] A. Lascoux, B. Leclerc, and J.Y. Thibon. The Plactic Monoid. Survey article available at [http://www-igm.univ-mlv.fr/~jyt/ARTICLES/plactic.ps]

TESTS:

sage: Word([1,1,2,2,3]).charge()
4
sage: Word([3,1,1,2,2]).charge()
3
sage: Word([2,1,1,2,3]).charge()
2
sage: Word([2,1,1,3,2]).charge()
2
sage: Word([3,2,1,1,2]).charge()
1
sage: Word([2,2,1,1,3]).charge()
1
sage: Word([3,2,2,1,1]).charge()
0
sage: Word([]).charge()
0

cocharge()

Returns the cocharge of self. For a word $$w$$, this can be defined as $$n_{ev} - ch(w)$$, where $$ch(w)$$ is the charge of $$w$$ and $$ev$$ is the evaluation of $$w$$, and $$n_{ev}$$ is $$\sum_{i<j} min(ev_i, ev_j)$$.

EXAMPLES:

sage: Word([1,2,3]).cocharge()
0
sage: Word([3,2,1]).cocharge()
3
sage: Word([1,1,2]).cocharge()
0
sage: Word([2,1,2]).cocharge()
1


TESTS:

sage: Word([]).cocharge()
0

coerce(other)

Tries to return a pair of words with a common parent; raises an exception if this is not possible.

This function begins by checking if both words have the same parent. If this is the case, then no work is done and both words are returned as-is.

Otherwise it will attempt to convert other to the domain of self. If that fails, it will attempt to convert self to the domain of other. If both attempts fail, it raises a TypeError to signal failure.

EXAMPLES:

sage: W1 = Words('abc'); W2 = Words('ab')
sage: w1 = W1('abc'); w2 = W2('abba'); w3 = W1('baab')
sage: w1.parent() is w2.parent()
False
sage: a, b = w1.coerce(w2)
sage: a.parent() is b.parent()
True
sage: w1.parent() is w2.parent()
False

colored_vector(x=0, y=0, width='default', height=1, cmap='hsv', thickness=1, label=None)

Returns a vector (Graphics object) illustrating self. Each letter is represented by a coloured rectangle.

If the parent of self is a class of words over a finite alphabet, then each letter in the alphabet is assigned a unique colour, and this colour will be the same every time this method is called. This is especially useful when plotting and comparing words defined on the same alphabet.

If the alphabet is infinite, then the letters appearing in the word are used as the alphabet.

INPUT:

• x - (default: 0) bottom left x-coordinate of the vector

• y - (default: 0) bottom left y-coordinate of the vector

• width - (default: ‘default’) width of the vector. By default, the width is the length of self.

• height - (default: 1) height of the vector

• thickness - (default: 1) thickness of the contour

• cmap - (default: ‘hsv’) color map; for available color map names

• label - str (default: None) a label to add on the colored vector.

OUTPUT:

Graphics

EXAMPLES:

sage: Word(range(20)).colored_vector()
sage: Word(range(100)).colored_vector(0,0,10,1)
sage: Words(range(100))(range(10)).colored_vector()
sage: w = Word('abbabaab')
sage: w.colored_vector()
sage: w.colored_vector(cmap='autumn')
sage: Word(range(20)).colored_vector(label='Rainbow')


When two words are defined under the same parent, same letters are mapped to same colors:

sage: W = Words(range(20))
sage: w = W(range(20))
sage: y = W(range(10,20))
sage: y.colored_vector(y=1, x=10) + w.colored_vector()


TESTS:

The empty word:

sage: Word().colored_vector()
sage: Word().colored_vector(label='empty')


Unknown cmap:

sage: Word(range(100)).colored_vector(cmap='jolies')
Traceback (most recent call last):
...
RuntimeError: Color map jolies not known
sage: Word(range(100)).colored_vector(cmap='__doc__')
Traceback (most recent call last):
...
RuntimeError: Color map __doc__ not known

commutes_with(other)

Returns True if self commutes with other, and False otherwise.

EXAMPLES:

sage: Word('12').commutes_with(Word('12'))
True
sage: Word('12').commutes_with(Word('11'))
False
sage: Word().commutes_with(Word('21'))
True

complete_return_words(fact)

Returns the set of complete return words of fact in self.

This is the set of all factors starting by the given factor and ending just after the next occurrence of this factor. See for instance [1].

INPUT:

• fact - a non empty finite word

OUTPUT:

Python set of finite words

EXAMPLES:

sage: s = Word('21331233213231').complete_return_words(Word('2'))
sage: sorted(s)
[word: 2132, word: 213312, word: 2332]
sage: Word('').complete_return_words(Word('213'))
set([])
sage: Word('121212').complete_return_words(Word('1212'))
set([word: 121212])


REFERENCES:

• [1] J. Justin, L. Vuillon, Return words in Sturmian and episturmian words, Theor. Inform. Appl. 34 (2000) 343–356.
concatenate(other)

Returns the concatenation of self and other.

INPUT:

• other - a word over the same alphabet as self

EXAMPLES:

Concatenation may be made using + or * operations:

sage: w = Word('abadafd')
sage: y = Word([5,3,5,8,7])
sage: w * y
sage: w + y
sage: w.concatenate(y)


Both words must be defined over the same alphabet:

sage: z = Word('12223', alphabet = '123')
sage: z + y
Traceback (most recent call last):
...
ValueError: 5 not in alphabet!


Eventually, it should work:

sage: z = Word('12223', alphabet = '123')
sage: z + y                   #todo: not implemented
word: 1222353587


TESTS:

The empty word is not considered by concatenation:

sage: type(Word([]) * Word('abcd'))
<class 'sage.combinat.words.word.FiniteWord_str'>
sage: type(Word('abcd') * Word())
<class 'sage.combinat.words.word.FiniteWord_str'>
sage: type(Word('abcd') * Word([]))
<class 'sage.combinat.words.word.FiniteWord_str'>
sage: type(Word('abcd') * Word(()))
<class 'sage.combinat.words.word.FiniteWord_str'>
sage: type(Word([1,2,3]) * Word(''))
<class 'sage.combinat.words.word.FiniteWord_list'>

conjugate(pos)

Returns the conjugate at pos of self.

pos can be any integer, the distance used is the modulo by the length of self.

EXAMPLES:

sage: Word('12112').conjugate(1)
word: 21121
sage: Word().conjugate(2)
word:
sage: Word('12112').conjugate(8)
word: 12121
sage: Word('12112').conjugate(-1)
word: 21211

conjugate_position(other)

Returns the position where self is conjugate with other. Returns None if there is no such position.

EXAMPLES:

sage: Word('12113').conjugate_position(Word('31211'))
1
sage: Word('12131').conjugate_position(Word('12113')) is None
True
sage: Word().conjugate_position(Word('123')) is None
True


TESTS:

We check that trac #11128 is fixed:

sage: w = Word([0,0,1,0,2,1])
sage: [w.conjugate(i).conjugate_position(w) for i in range(w.length())]
[0, 1, 2, 3, 4, 5]

conjugates()

Returns the list of unique conjugates of self.

EXAMPLES:

sage: Word(range(6)).conjugates()
[word: 012345,
word: 123450,
word: 234501,
word: 345012,
word: 450123,
word: 501234]
sage: Word('cbbca').conjugates()
[word: cbbca, word: bbcac, word: bcacb, word: cacbb, word: acbbc]


The result contains each conjugate only once:

sage: Word('abcabc').conjugates()
[word: abcabc, word: bcabca, word: cabcab]


TESTS:

sage: Word().conjugates()
[word: ]
sage: Word('a').conjugates()
[word: a]

conjugates_iterator()

Returns an iterator over the conjugates of self.

EXAMPLES:

sage: it = Word(range(4)).conjugates_iterator()
sage: for w in it: w
word: 0123
word: 1230
word: 2301
word: 3012

count(letter)

Counts the number of occurrences of letter in self.

EXAMPLES:

sage: Word('abbabaab').count('a')
4

critical_exponent()

Returns the critical exponent of self.

The critical exponent of a word is the supremum of the order of all its (finite) factors. See [1].

Note

The implementation here uses the suffix tree to enumerate all the factors. It should be improved.

EXAMPLES:

sage: Word('aaba').critical_exponent()
2
sage: Word('aabaa').critical_exponent()
2
sage: Word('aabaaba').critical_exponent()
7/3
sage: Word('ab').critical_exponent()
1
sage: Word('aba').critical_exponent()
3/2
sage: words.ThueMorseWord()[:20].critical_exponent()
2


REFERENCES:

• [1] F. Dejean. Sur un théorème de Thue. J. Combinatorial Theory Ser. A 13:90–99, 1972.
crochemore_factorization()

Returns the Crochemore factorization of self as an ordered list of factors.

The Crochemore factorization of a finite word $$w$$ is the unique factorization: $$(x_1, x_2, \ldots, x_n)$$ of $$w$$ with each $$x_i$$ satisfying either: C1. $$x_i$$ is a letter that does not appear in $$u = x_1\ldots x_{i-1}$$; C2. $$x_i$$ is the longest prefix of $$v = x_i\ldots x_n$$ that also has an occurrence beginning within $$u = x_1\ldots x_{i-1}$$. See [1].

Note

This is not a very good implementation, and should be improved.

EXAMPLES:

sage: x = Word('abababb')
sage: x.crochemore_factorization()
(a, b, abab, b)
sage: mul(x.crochemore_factorization()) == x
True
sage: y = Word('abaababacabba')
sage: y.crochemore_factorization()
(a, b, a, aba, ba, c, ab, ba)
sage: mul(y.crochemore_factorization()) == y
True
sage: x = Word([0,1,0,1,0,1,1])
sage: x.crochemore_factorization()
(0, 1, 0101, 1)
sage: mul(x.crochemore_factorization()) == x
True


REFERENCES:

• [1] M. Crochemore, Recherche linéaire d’un carré dans un mot, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 14 781–784.
defect(f=None)

Returns the defect of self.

The defect of a finite word $$w$$ is given by the difference between the maximum number of possible palindromic factors in a word of length $$|w|$$ and the actual number of palindromic factors contained in $$w$$. It is well known that the maximum number of palindromic factors in $$w$$ is $$|w|+1$$ (see [DJP01]).

An optional involution on letters f can be given. In that case, the f-palindromic defect (or pseudopalindromic defect, or theta-palindromic defect) of $$w$$ is returned. It is a generalization of defect to f-palindromes. More precisely, the defect is $$D(w)=|w|+1-g_f(w)-|PAL_f(w)|$$, where $$PAL_f(w)$$ denotes the set of f-palindromic factors of $$w$$ (including the empty word) and $$g_f(w)$$ is the number of pairs $$\{a, f(a)\}$$ such that $$a$$ is a letter, $$a$$ is not equal to $$f(a)$$, and $$a$$ or $$f(a)$$ occurs in $$w$$. In the case of usual palindromes (i.e., for f not given or equal to the identity), $$g_f(w) = 0$$ for all $$w$$. See [BHNR04] for usual palindromes and [Sta11] for f-palindromes.

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e., $$f$$ equal to the identity.

OUTPUT:

integer – If f is None, the palindromic defect of self;
otherwise, the f-palindromic defect of self.

EXAMPLES:

sage: Word('ara').defect()
0
sage: Word('abcacba').defect()
1


It is known that Sturmian words (see [DJP01]) have zero defect:

sage: words.FibonacciWord()[:100].defect()
0

sage: sa = WordMorphism('a->ab,b->b')
sage: sb = WordMorphism('a->a,b->ba')
sage: w = (sa*sb*sb*sa*sa*sa*sb).fixed_point('a')
sage: w[:30].defect()
0
sage: w[110:140].defect()
0


It is even conjectured that the defect of an aperiodic word which is a fixed point of a primitive morphism is either $$0$$ or infinite (see [BBGL08]):

sage: w = words.ThueMorseWord()
sage: w[:50].defect()
12
sage: w[:100].defect()
16
sage: w[:300].defect()
52


For generalized defect with an involution different from the identity, there is always a letter which is not a palindrome! This is the reason for the modification of the definition:

sage: f = WordMorphism('a->b,b->a')
sage: Word('a').defect(f)
0
sage: Word('ab').defect(f)
0
sage: Word('aa').defect(f)
1
sage: Word('abbabaabbaababba').defect(f)
3

sage: f = WordMorphism('a->b,b->a,c->c')
sage: Word('cabc').defect(f)
0
sage: Word('abcaab').defect(f)
2


Other examples:

sage: Word('000000000000').defect()
0
sage: Word('011010011001').defect()
2
sage: Word('0101001010001').defect()
0
sage: Word().defect()
0
sage: Word('abbabaabbaababba').defect()
2


REFERENCES:

 [BBGL08] A. Blondin Massé, S. Brlek, A. Garon, and S. Labbé, Combinatorial properties of f -palindromes in the Thue-Morse sequence. Pure Math. Appl., 19(2-3):39–52, 2008.
 [BHNR04] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
 [DJP01] (1, 2) X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255, (2001), no. 1–2, 539–553.
 [Sta11] Š. Starosta, On Theta-palindromic Richness, Theoret. Comp. Sci. 412 (2011) 1111–1121
deg_inv_lex_less(other, weights=None)

Returns True if the word self is degree inverse lexicographically less than other.

EXAMPLES:

sage: Word([1,2,4]).deg_inv_lex_less(Word([1,3,2]))
False
sage: Word([3,2,1]).deg_inv_lex_less(Word([1,2,3]))
True

deg_lex_less(other, weights=None)

Returns True if self is degree lexicographically less than other, and False otherwise. The weight of each letter in the ordered alphabet is given by weights, which defaults to [1, 2, 3, ...].

EXAMPLES:

sage: Word([1,2,3]).deg_lex_less(Word([1,3,2]))
True
sage: Word([3,2,1]).deg_lex_less(Word([1,2,3]))
False
sage: W = Words(range(5))
sage: W([1,2,4]).deg_lex_less(W([1,3,2]))
False
sage: Word("abba").deg_lex_less(Word("abbb"), dict(a=1,b=2))
True
sage: Word("abba").deg_lex_less(Word("baba"), dict(a=1,b=2))
True
sage: Word("abba").deg_lex_less(Word("aaba"), dict(a=1,b=2))
False
sage: Word("abba").deg_lex_less(Word("aaba"), dict(a=1,b=0))
True

deg_rev_lex_less(other, weights=None)

Returns True if self is degree reverse lexicographically less than other.

EXAMPLES:

sage: Word([3,2,1]).deg_rev_lex_less(Word([1,2,3]))
False
sage: Word([1,2,4]).deg_rev_lex_less(Word([1,3,2]))
False
sage: Word([1,2,3]).deg_rev_lex_less(Word([1,2,4]))
True

degree(weights=None)

Returns the weighted degree of self, where the weighted degree of each letter in the ordered alphabet is given by weights, which defaults to [1, 2, 3, ...].

INPUTS:

• weights - a list or tuple, or a dictionary keyed by the letters occurring in self.

EXAMPLES:

sage: Word([1,2,3]).degree()
6
sage: Word([3,2,1]).degree()
6
sage: Words("ab")("abba").degree()
6
sage: Words("ab")("abba").degree([0,2])
4
sage: Words("ab")("abba").degree([-1,-1])
-4
sage: Words("ab")("aabba").degree([1,1])
5
sage: Words([1,2,4])([1,2,4]).degree()
6
sage: Word([1,2,4]).degree()
7
sage: Word("aabba").degree({'a':1,'b':2})
7
sage: Word([0,1,0]).degree({0:17,1:0})
34

delta()

Returns the image of self under the delta morphism. This is the word composed of the length of consecutive runs of the same letter in a given word.

EXAMPLES:

sage: W = Words('0123456789')
sage: W('22112122').delta()
word: 22112
sage: W('555008').delta()
word: 321
sage: W().delta()
word:
sage: Word('aabbabaa').delta()
word: 22112

delta_derivate(W=None)

Returns the derivative under delta for self.

EXAMPLES:

sage: W = Words('12')
sage: W('12211').delta_derivate()
word: 22
sage: W('1').delta_derivate(Words([1]))
word: 1
sage: W('2112').delta_derivate()
word: 2
sage: W('2211').delta_derivate()
word: 22
sage: W('112').delta_derivate()
word: 2
sage: W('11222').delta_derivate(Words([1, 2, 3]))
word: 3

delta_derivate_left(W=None)

Returns the derivative under delta for self.

EXAMPLES:

sage: W = Words('12')
sage: W('12211').delta_derivate_left()
word: 22
sage: W('1').delta_derivate_left(Words([1]))
word: 1
sage: W('2112').delta_derivate_left()
word: 21
sage: W('2211').delta_derivate_left()
word: 22
sage: W('112').delta_derivate_left()
word: 21
sage: W('11222').delta_derivate_left(Words([1, 2, 3]))
word: 3

delta_derivate_right(W=None)

Returns the right derivative under delta for self.

EXAMPLES:

sage: W = Words('12')
sage: W('12211').delta_derivate_right()
word: 122
sage: W('1').delta_derivate_right(Words([1]))
word: 1
sage: W('2112').delta_derivate_right()
word: 12
sage: W('2211').delta_derivate_right()
word: 22
sage: W('112').delta_derivate_right()
word: 2
sage: W('11222').delta_derivate_right(Words([1, 2, 3]))
word: 23

delta_inv(W=None, s=None)

Lifts self via the delta operator to obtain a word containing the letters in alphabet (default is [0, 1]). The letters used in the construction start with s (default is alphabet[0]) and cycle through alphabet.

INPUT:

• alphabet - an iterable
• s - an object in the iterable

EXAMPLES:

sage: W = Words([1, 2])
sage: W([2, 2, 1, 1]).delta_inv()
word: 112212
sage: W([1, 1, 1, 1]).delta_inv(Words('123'))
word: 1231
sage: W([2, 2, 1, 1, 2]).delta_inv(s=2)
word: 22112122

evaluation(alphabet=None)

Returns the Parikh vector of self, i.e., the vector containing the number of occurrences of each letter, given in the order of the alphabet.

INPUT:

• alphabet - (default: None) finite ordered alphabet, if None it uses the set of letters in self with the ordering defined by the parent

EXAMPLES:

sage: Words('ab')().parikh_vector()
[0, 0]
sage: Word('aabaa').parikh_vector('abc')
[4, 1, 0]
sage: Word('a').parikh_vector('abc')
[1, 0, 0]
sage: Word('a').parikh_vector('cab')
[0, 1, 0]
sage: Word('a').parikh_vector('bca')
[0, 0, 1]
sage: Word().parikh_vector('ab')
[0, 0]
sage: Word().parikh_vector('abc')
[0, 0, 0]
sage: Word().parikh_vector('abcd')
[0, 0, 0, 0]


TESTS:

sage: Word('aabaa').parikh_vector()
Traceback (most recent call last):
...
TypeError: the alphabet is infinite; specify a finite alphabet or use evaluation_dict() instead

evaluation_dict()

Returns a dictionary keyed by the letters occurring in self with values the number of occurrences of the letter.

EXAMPLES:

sage: Word([2,1,4,2,3,4,2]).evaluation_dict()
{1: 1, 2: 3, 3: 1, 4: 2}
{'a': 1, 'c': 1, 'b': 3, 'd': 2}
sage: Word().evaluation_dict()
{}

evaluation_partition()

Returns the evaluation of the word w as a partition.

EXAMPLES:

sage: Word("acdabda").evaluation_partition()
[3, 2, 1, 1]
sage: Word([2,1,4,2,3,4,2]).evaluation_partition()
[3, 2, 1, 1]

evaluation_sparse()

Returns a list representing the evaluation of self. The entries of the list are two-element lists [a, n], where a is a letter occurring in self and n is the number of occurrences of a in self.

EXAMPLES:

sage: Word([4,4,2,5,2,1,4,1]).evaluation_sparse()
[(1, 2), (2, 2), (4, 3), (5, 1)]
sage: Word("abcaccab").evaluation_sparse()
[('a', 3), ('c', 3), ('b', 2)]

exponent()

Returns the exponent of self.

OUTPUT:

integer – the exponent

EXAMPLES:

sage: Word('1231').exponent()
1
sage: Word('121212').exponent()
3
sage: Word().exponent()
0

factor_iterator(n=None)

Generates distinct factors of self.

INPUT:

• n - an integer, or None.

OUTPUT:

If n is an integer, returns an iterator over all distinct factors of length n. If n is None, returns an iterator generating all distinct factors.

EXAMPLES:

sage: w = Word('1213121')
sage: sorted( w.factor_iterator(0) )
[word: ]
sage: sorted( w.factor_iterator(10) )
[]
sage: sorted( w.factor_iterator(1) )
[word: 1, word: 2, word: 3]
sage: sorted( w.factor_iterator(4) )
[word: 1213, word: 1312, word: 2131, word: 3121]
sage: sorted( w.factor_iterator() )
[word: , word: 1, word: 12, word: 121, word: 1213, word: 12131, word: 121312, word: 1213121, word: 13, word: 131, word: 1312, word: 13121, word: 2, word: 21, word: 213, word: 2131, word: 21312, word: 213121, word: 3, word: 31, word: 312, word: 3121]

sage: u = Word([1,2,1,2,3])
sage: sorted( u.factor_iterator(0) )
[word: ]
sage: sorted( u.factor_iterator(10) )
[]
sage: sorted( u.factor_iterator(1) )
[word: 1, word: 2, word: 3]
sage: sorted( u.factor_iterator(5) )
[word: 12123]
sage: sorted( u.factor_iterator() )
[word: , word: 1, word: 12, word: 121, word: 1212, word: 12123, word: 123, word: 2, word: 21, word: 212, word: 2123, word: 23, word: 3]

sage: xxx = Word("xxx")
sage: sorted( xxx.factor_iterator(0) )
[word: ]
sage: sorted( xxx.factor_iterator(4) )
[]
sage: sorted( xxx.factor_iterator(2) )
[word: xx]
sage: sorted( xxx.factor_iterator() )
[word: , word: x, word: xx, word: xxx]

sage: e = Word()
sage: sorted( e.factor_iterator(0) )
[word: ]
sage: sorted( e.factor_iterator(17) )
[]
sage: sorted( e.factor_iterator() )
[word: ]


TESTS:

sage: type( Word('cacao').factor_iterator() )
<type 'generator'>

factor_occurrences_in(other)

Returns an iterator over all occurrences (including overlapping ones) of self in other in their order of appearance.

EXAMPLES:

sage: u = Word('121')
sage: w = Word('121213211213')
sage: list(u.factor_occurrences_in(w))
[0, 2, 8]

factor_set(n=None)

Returns the set of factors (of length n) of self.

INPUT:

• n - an integer or None (default: None).

OUTPUT:

If n is an integer, returns the set of all distinct factors of length n. If n is None, returns the set of all distinct factors.

EXAMPLES:

sage: w = Word('121')
sage: s = w.factor_set()
sage: sorted(s)
[word: , word: 1, word: 12, word: 121, word: 2, word: 21]

sage: w = Word('1213121')
sage: for i in range(w.length()): sorted(w.factor_set(i))
[word: ]
[word: 1, word: 2, word: 3]
[word: 12, word: 13, word: 21, word: 31]
[word: 121, word: 131, word: 213, word: 312]
[word: 1213, word: 1312, word: 2131, word: 3121]
[word: 12131, word: 13121, word: 21312]
[word: 121312, word: 213121]

sage: w = Word([1,2,1,2,3])
sage: s = w.factor_set()
sage: sorted(s)
[word: , word: 1, word: 12, word: 121, word: 1212, word: 12123, word: 123, word: 2, word: 21, word: 212, word: 2123, word: 23, word: 3]


TESTS:

sage: w = Word("xx")
sage: s = w.factor_set()
sage: sorted(s)
[word: , word: x, word: xx]

sage: Set(Word().factor_set())
{word: }

find(sub, start=0, end=None)

Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure.

INPUT:

• sub - string or word to search for.
• start - non negative integer (default: 0) specifying the position from which to start the search.
• end - non negative integer (default: None) specifying the position at which the search must stop. If None, then the search is performed up to the end of the string.

OUTPUT:

non negative integer or -1

EXAMPLES:

sage: w = Word([0,1,0,0,1])
sage: w.find(Word([0,1]))
0
sage: w.find(Word([0,1]), start=1)
3
sage: w.find(Word([0,1]), start=1, end=5)
3
sage: w.find(Word([0,1]), start=1, end=4) == -1
True
sage: w.find(Word([1,1])) == -1
True


Instances of Word_str handle string inputs as well:

sage: w = Word('abac')
sage: w.find('a')
0
sage: w.find(Word('a'))
0

first_pos_in(other)

Returns the position of the first occurrence of self in other, or None if self is not a factor of other.

EXAMPLES:

sage: Word('12').first_pos_in(Word('131231'))
2
sage: Word('32').first_pos_in(Word('131231')) is None
True

freq()

Returns a table of the frequencies of the letters in self.

OUTPUT:

dict – letters associated to their frequency

EXAMPLES:

sage: f = Word('1213121').freq()
doctest:1: DeprecationWarning: freq is deprecated, use evaluation_dict instead!
See http://trac.sagemath.org/8429 for details.
sage: f # keys appear in random order
{'1': 4, '2': 2, '3': 1}


TESTS:

sage: f = Word('1213121').freq()
sage: f['1'] == 4
True
sage: f['2'] == 2
True
sage: f['3'] == 1
True

good_suffix_table()

Returns a table of the maximum skip you can do in order not to miss a possible occurrence of self in a word.

This is a part of the Boyer-Moore algorithm to find factors. See [1].

EXAMPLES:

sage: Word('121321').good_suffix_table()
[5, 5, 5, 5, 3, 3, 1]
sage: Word('12412').good_suffix_table()
[3, 3, 3, 3, 3, 1]


REFERENCES:

• [1] R.S. Boyer, J.S. Moore, A fast string searching algorithm, Communications of the ACM 20 (1977) 762–772.
has_period(p)

Returns True if self has the period p, False otherwise.

Note

By convention, integers greater than the length of self are periods of self.

INPUT:

• p - an integer to check if it is a period of self.

EXAMPLES:

sage: w = Word('ababa')
sage: w.has_period(2)
True
sage: w.has_period(3)
False
sage: w.has_period(4)
True
sage: w.has_period(-1)
False
sage: w.has_period(5)
True
sage: w.has_period(6)
True

has_prefix(other)

Test whether self has other as a prefix.

INPUT:

• other - a word, or data describing a word

OUTPUT:

• boolean

EXAMPLES:

sage: w = Word("abbabaabababa")
sage: u = Word("abbab")
sage: w.has_prefix(u)
True
sage: u.has_prefix(w)
False
sage: u.has_prefix("abbab")
True

sage: w = Word([0,1,1,0,1,0,0,1,0,1,0,1,0])
sage: u = Word([0,1,1,0,1])
sage: w.has_prefix(u)
True
sage: u.has_prefix(w)
False
sage: u.has_prefix([0,1,1,0,1])
True

has_suffix(other)

Test whether self has other as a suffix.

Note

Some word datatype classes, like WordDatatype_str, override this method.

INPUT:

• other - a word, or data describing a word

OUTPUT:

• boolean

EXAMPLES:

sage: w = Word("abbabaabababa")
sage: u = Word("ababa")
sage: w.has_suffix(u)
True
sage: u.has_suffix(w)
False
sage: u.has_suffix("ababa")
True

sage: w = Word([0,1,1,0,1,0,0,1,0,1,0,1,0])
sage: u = Word([0,1,0,1,0])
sage: w.has_suffix(u)
True
sage: u.has_suffix(w)
False
sage: u.has_suffix([0,1,0,1,0])
True

implicit_suffix_tree()

Returns the implicit suffix tree of self.

The suffix tree of a word $$w$$ is a compactification of the suffix trie for $$w$$. The compactification removes all nodes that have exactly one incoming edge and exactly one outgoing edge. It consists of two components: a tree and a word. Thus, instead of labelling the edges by factors of $$w$$, we can labelled them by indices of the occurrence of the factors in $$w$$.

EXAMPLES:

sage: w = Word("cacao")
sage: w.implicit_suffix_tree()
Implicit Suffix Tree of the word: cacao

sage: w = Word([0,1,0,1,1])
sage: w.implicit_suffix_tree()
Implicit Suffix Tree of the word: 01011

inv_lex_less(other)

Returns True if self is inverse lexicographically less than other.

EXAMPLES:

sage: Word([1,2,4]).inv_lex_less(Word([1,3,2]))
False
sage: Word([3,2,1]).inv_lex_less(Word([1,2,3]))
True

inversions()

Returns a list of the inversions of self. An inversion is a pair (i,j) of non-negative integers i < j such that self[i] > self[j].

EXAMPLES:

sage: Word([1,2,3,2,2,1]).inversions()
[[1, 5], [2, 3], [2, 4], [2, 5], [3, 5], [4, 5]]
sage: Words([3,2,1])([1,2,3,2,2,1]).inversions()
[[0, 1], [0, 2], [0, 3], [0, 4], [1, 2]]
sage: Word('abbaba').inversions()
[[1, 3], [1, 5], [2, 3], [2, 5], [4, 5]]
sage: Words('ba')('abbaba').inversions()
[[0, 1], [0, 2], [0, 4], [3, 4]]

is_balanced(q=1)

Returns True if self is $$q$$-balanced, and False otherwise.

A finite or infinite word $$w$$ is said to be $$q$$-balanced if for any two factors $$u$$, $$v$$ of $$w$$ of the same length, the difference between the number of $$x$$‘s in each of $$u$$ and $$v$$ is at most $$q$$ for all letters $$x$$ in the alphabet of $$w$$. A $$1$$-balanced word is simply said to be balanced. See for instance [1] and Chapter 2 of [2].

INPUT:

• q - integer (default 1), the balance level

OUTPUT:

boolean – the result

EXAMPLES:

sage: Word('1213121').is_balanced()
True
sage: Word('1122').is_balanced()
False
sage: Word('121333121').is_balanced()
False
sage: Word('121333121').is_balanced(2)
False
sage: Word('121333121').is_balanced(3)
True
sage: Word('121122121').is_balanced()
False
sage: Word('121122121').is_balanced(2)
True


TESTS:

sage: Word('121122121').is_balanced(-1)
Traceback (most recent call last):
...
TypeError: the balance level must be a positive integer
sage: Word('121122121').is_balanced(0)
Traceback (most recent call last):
...
TypeError: the balance level must be a positive integer
sage: Word('121122121').is_balanced('a')
Traceback (most recent call last):
...
TypeError: the balance level must be a positive integer


REFERENCES:

• [1] J. Cassaigne, S. Ferenczi, L.Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble) 50 (2000) 1265–1276.
• [2] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.

Returns True if seq is a cadence of self, and False otherwise.

A cadence is an increasing sequence of indexes that all map to the same letter.

EXAMPLES:

sage: Word('121132123').is_cadence([0, 2, 6])
True
False
True

is_conjugate_with(other)

Returns True if self is a conjugate of other, and False otherwise.

INPUT:

• other - a finite word

OUPUT

bool

EXAMPLES:

sage: w = Word([0..20])
sage: z = Word([7..20] + [0..6])
sage: w
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20
sage: z
word: 7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6
sage: w.is_conjugate_with(z)
True
sage: z.is_conjugate_with(w)
True
sage: u = Word([4]*21)
sage: u.is_conjugate_with(w)
False
sage: u.is_conjugate_with(z)
False


Both words must be finite:

sage: w = Word(iter([2]*100),length='unknown')
sage: z = Word([2]*100)
sage: z.is_conjugate_with(w) #TODO: Not implemented for word of unknown length
True
sage: wf = Word(iter([2]*100),length='finite')
sage: z.is_conjugate_with(wf)
True
sage: wf.is_conjugate_with(z)
True


TESTS:

sage: Word('11213').is_conjugate_with(Word('31121'))
True
sage: Word().is_conjugate_with(Word('123'))
False
sage: Word('112131').is_conjugate_with(Word('11213'))
False
sage: Word('12131').is_conjugate_with(Word('11213'))
True


We make sure that trac #11128 is fixed:

sage: Word('abaa').is_conjugate_with(Word('aaba'))
True
sage: Word('aaba').is_conjugate_with(Word('abaa'))
True

is_cube()

Returns True if self is a cube, and False otherwise.

EXAMPLES:

sage: Word('012012012').is_cube()
True
sage: Word('01010101').is_cube()
False
sage: Word().is_cube()
True
sage: Word('012012').is_cube()
False

is_cube_free()

Returns True if self does not contain cubes, and False otherwise.

EXAMPLES:

sage: Word('12312').is_cube_free()
True
sage: Word('32221').is_cube_free()
False
sage: Word().is_cube_free()
True


TESTS:

We make sure that #8490 is fixed:

sage: Word('111').is_cube_free()
False
sage: Word('2111').is_cube_free()
False
sage: Word('32111').is_cube_free()
False

is_empty()

Returns True if the length of self is zero, and False otherwise.

EXAMPLES:

sage: Word([]).is_empty()
True
sage: Word('a').is_empty()
False

is_factor(other)

Returns True if self is a factor of other, and False otherwise.

EXAMPLES:

sage: u = Word('2113')
sage: w = Word('123121332131233121132123')
sage: u.is_factor(w)
True
sage: u = Word('321')
sage: w = Word('1231241231312312312')
sage: u.is_factor(w)
False


The empty word is factor of another word:

sage: Word().is_factor(Word())
True
sage: Word().is_factor(Word('a'))
True
sage: Word().is_factor(Word([1,2,3]))
True
sage: Word().is_factor(Word(lambda n:n, length=5))
True

is_factor_of(other)

Returns True if self is a factor of other, and False otherwise.

EXAMPLES:

sage: u = Word('2113')
sage: w = Word('123121332131233121132123')
sage: u.is_factor_of(w)
doctest:1: DeprecationWarning: is_factor_of is deprecated, use is_factor instead!
See http://trac.sagemath.org/8429 for details.
True
sage: u = Word('321')
sage: w = Word('1231241231312312312')
sage: u.is_factor_of(w)
False

is_finite()

Returns True.

EXAMPLES:

sage: Word([]).is_finite()
True
sage: Word('a').is_finite()
True

is_full(f=None)

Returns True if self has defect 0, and False otherwise.

A word is full (or rich) if its defect is zero (see [1]). If f is given, then the f-palindromic defect is used (see [2]).

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

boolean – If f is None, whether self is full;
otherwise, whether self is full of $$f$$-palindromes.

EXAMPLES:

sage: words.ThueMorseWord()[:100].is_full()
False
sage: words.FibonacciWord()[:100].is_full()
True
sage: Word('000000000000000').is_full()
True
sage: Word('011010011001').is_full()
False
sage: Word('2194').is_full()
True
sage: Word().is_full()
True

sage: f = WordMorphism('a->b,b->a')
sage: Word().is_full(f)
True
sage: w = Word('ab')
sage: w.is_full()
True
sage: w.is_full(f)
True

sage: f = WordMorphism('a->b,b->a')
sage: Word('abab').is_full(f)
True
sage: Word('abba').is_full(f)
False


A simple example of an infinite word full of f-palindromes:

sage: p = WordMorphism({0:'abc',1:'ab'})
sage: f = WordMorphism('a->b,b->a,c->c')
sage: p(words.FibonacciWord()[:50]).is_full(f)
True
sage: p(words.FibonacciWord()[:150]).is_full(f)
True


REFERENCES:

• [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
• [2] E. Pelantová, Š. Starosta, Infinite words rich and almost rich in generalized palindromes, in: G. Mauri, A. Leporati (Eds.), Developments in Language Theory, volume 6795 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 406–416
is_lyndon()

Returns True if self is a Lyndon word, and False otherwise.

A Lyndon word is a non-empty word that is lexicographically smaller than all of its proper suffixes for the given order on its alphabet. That is, $$w$$ is a Lyndon word if $$w$$ is non-empty and for each factorization $$w = uv$$ (with $$u$$, $$v$$ both non-empty), we have $$w < v$$.

Equivalently, $$w$$ is a Lyndon word iff $$w$$ is a non-empty word that is lexicographically smaller than all of its proper conjugates for the given order on its alphabet.

See for instance [1].

EXAMPLES:

sage: Word('123132133').is_lyndon()
True
sage: Word().is_lyndon()
True
sage: Word('122112').is_lyndon()
False


TESTS:

A sanity check: LyndonWords generates Lyndon words, so we filter all words of length $$n<10$$ on the alphabet [1,2,3] for Lyndon words, and compare with the LyndonWords generator:

sage: for n in range(1,10):
...       lw1 = [w for w in Words([1,2,3], n) if w.is_lyndon()]
...       lw2 = LyndonWords(3,n)
...       if set(lw1) != set(lw2): print False


Filter all words of length 8 on the alphabet [c,a,b] for Lyndon words, and compare with the LyndonWords generator after mapping [a,b,c] to [2,3,1]:

sage: lw = [w for w in Words('cab', 8) if w.is_lyndon()]
sage: phi = WordMorphism({'a':2,'b':3,'c':1})
sage: set(map(phi, lw)) == set(LyndonWords(3,8))
True


REFERENCES:

• [1] M. Lothaire, Combinatorics On Words, vol. 17 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Massachusetts, 1983.
is_overlap()

Returns True if self is an overlap, and False otherwise.

EXAMPLES:

sage: Word('12121').is_overlap()
True
sage: Word('123').is_overlap()
False
sage: Word('1231').is_overlap()
False
sage: Word('123123').is_overlap()
False
sage: Word('1231231').is_overlap()
True
sage: Word().is_overlap()
False

is_palindrome(f=None)

Returns True if self is a palindrome (or a $$f$$-palindrome), and False otherwise.

Let $$f : \Sigma \rightarrow \Sigma$$ be an involution that extends to a morphism on $$\Sigma^*$$. We say that $$w\in\Sigma^*$$ is a f-palindrome if $$w=f(\tilde{w})$$ [1]. Also called f-pseudo-palindrome [2].

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e., $$f$$ equal to the identity.

EXAMPLES:

sage: Word('esope reste ici et se repose').is_palindrome()
False
sage: Word('esoperesteicietserepose').is_palindrome()
True
sage: Word('I saw I was I').is_palindrome()
True
sage: Word('abbcbba').is_palindrome()
True
sage: Word('abcbdba').is_palindrome()
False


Some $$f$$-palindromes:

sage: f = WordMorphism('a->b,b->a')
sage: Word('aababb').is_palindrome(f)
True

sage: f = WordMorphism('a->b,b->a,c->c')
sage: Word('abacbacbab').is_palindrome(f)
True

sage: f = WordMorphism({'a':'b','b':'a'})
sage: Word('aababb').is_palindrome(f)
True

sage: f = WordMorphism({0:[1],1:[0]})
sage: w = words.ThueMorseWord()[:8]; w
word: 01101001
sage: w.is_palindrome(f)
True


The word must be in the domain of the involution:

sage: f = WordMorphism('a->a')
sage: Word('aababb').is_palindrome(f)
Traceback (most recent call last):
...
KeyError: 'b'


TESTS:

If the given involution is not an involution:

sage: f = WordMorphism('a->b,b->b')
sage: Word('abab').is_palindrome(f)
Traceback (most recent call last):
...
TypeError: self (=a->b, b->b) is not an endomorphism

sage: Y = Word
sage: Y().is_palindrome()
True
sage: Y('a').is_palindrome()
True
sage: Y('ab').is_palindrome()
False
sage: Y('aba').is_palindrome()
True
sage: Y('aa').is_palindrome()
True
sage: E = WordMorphism('a->b,b->a')
sage: Y().is_palindrome(E)
True
sage: Y('a').is_palindrome(E)
False
sage: Y('ab').is_palindrome(E)
True
sage: Y('aa').is_palindrome(E)
False
sage: Y('aba').is_palindrome(E)
False
sage: Y('abab').is_palindrome(E)
True


REFERENCES:

• [1] S. Labbé, Propriétés combinatoires des $$f$$-palindromes, Mémoire de maîtrise en Mathématiques, Montréal, UQAM, 2008, 109 pages.
• [2] V. Anne, L.Q. Zamboni, I. Zorca, Palindromes and Pseudo- Palindromes in Episturmian and Pseudo-Palindromic Infinite Words, in : S. Brlek, C. Reutenauer (Eds.), Words 2005, Publications du LaCIM, Vol. 36 (2005) 91–100.
is_prefix(other)

Returns True if self is a prefix of other, and False otherwise.

EXAMPLES:

sage: w = Word('0123456789')
sage: y = Word('012345')
sage: y.is_prefix(w)
True
sage: w.is_prefix(y)
False
sage: w.is_prefix(Word())
False
sage: Word().is_prefix(w)
True
sage: Word().is_prefix(Word())
True

is_prefix_of(other)

Returns True if self is a prefix of other, and False otherwise.

EXAMPLES:

sage: w = Word('0123456789')
sage: y = Word('012345')
sage: y.is_prefix_of(w)
doctest:1: DeprecationWarning: is_prefix_of is deprecated, use is_prefix instead!
See http://trac.sagemath.org/8429 for details.
True
sage: w.is_prefix_of(y)
False
sage: w.is_prefix_of(Word())
False
sage: Word().is_prefix_of(w)
True
sage: Word().is_prefix_of(Word())
True

is_primitive()

Returns True if self is primitive, and False otherwise.

A finite word $$w$$ is primitive if it is not a positive integer power of a shorter word.

EXAMPLES:

sage: Word('1231').is_primitive()
True
sage: Word('111').is_primitive()
False

is_proper_prefix(other)

Returns True if self is a proper prefix of other, and False otherwise.

EXAMPLES:

sage: Word('12').is_proper_prefix(Word('123'))
True
sage: Word('12').is_proper_prefix(Word('12'))
False
sage: Word().is_proper_prefix(Word('123'))
True
sage: Word('123').is_proper_prefix(Word('12'))
False
sage: Word().is_proper_prefix(Word())
False

is_proper_prefix_of(other)

Returns True if self is a proper prefix of other, and False otherwise.

EXAMPLES:

sage: Word('12').is_proper_prefix_of(Word('123'))
doctest:1: DeprecationWarning: is_proper_prefix_of is deprecated, use is_proper_prefix instead!
See http://trac.sagemath.org/8429 for details.
doctest:...: DeprecationWarning: is_prefix_of is deprecated, use is_prefix instead!
See http://trac.sagemath.org/8429 for details.
True
sage: Word('12').is_proper_prefix_of(Word('12'))
False
sage: Word().is_proper_prefix_of(Word('123'))
True
sage: Word('123').is_proper_prefix_of(Word('12'))
False
sage: Word().is_proper_prefix_of(Word())
False

is_proper_suffix(other)

Returns True if self is a proper suffix of other, and False otherwise.

EXAMPLES:

sage: Word('23').is_proper_suffix(Word('123'))
True
sage: Word('12').is_proper_suffix(Word('12'))
False
sage: Word().is_proper_suffix(Word('123'))
True
sage: Word('123').is_proper_suffix(Word('12'))
False

is_proper_suffix_of(other)

Returns True if self is a proper suffix of other, and False otherwise.

EXAMPLES:

sage: Word('23').is_proper_suffix_of(Word('123'))
doctest:1: DeprecationWarning: is_proper_suffix_of is deprecated, use is_proper_suffix instead!
See http://trac.sagemath.org/8429 for details.
doctest:...: DeprecationWarning: is_suffix_of is deprecated, use is_suffix instead!
See http://trac.sagemath.org/8429 for details.
True
sage: Word('12').is_proper_suffix_of(Word('12'))
False
sage: Word().is_proper_suffix_of(Word('123'))
True
sage: Word('123').is_proper_suffix_of(Word('12'))
False

is_quasiperiodic()

Returns True if self is quasiperiodic, and False otherwise.

A finite or infinite word $$w$$ is quasiperiodic if it can be constructed by concatenations and superpositions of one of its proper factors $$u$$, which is called a quasiperiod of $$w$$. See for instance [1], [2], and [3].

EXAMPLES:

sage: Word('abaababaabaababaaba').is_quasiperiodic()
True
sage: Word('abacaba').is_quasiperiodic()
False
sage: Word('a').is_quasiperiodic()
False
sage: Word().is_quasiperiodic()
False
sage: Word('abaaba').is_quasiperiodic()
True


REFERENCES:

• [1] A. Apostolico, A. Ehrenfeucht, Efficient detection of quasiperiodicities in strings, Theoret. Comput. Sci. 119 (1993) 247–265.
• [2] S. Marcus, Quasiperiodic infinite words, Bull. Eur. Assoc. Theor. Comput. Sci. 82 (2004) 170-174.
• [3] A. Glen, F. Levé, G. Richomme, Quasiperiodic and Lyndon episturmian words, Preprint, 2008, arXiv:0805.0730.
is_rich(f=None)

Returns True if self has defect 0, and False otherwise.

A word is full (or rich) if its defect is zero (see [1]). If f is given, then the f-palindromic defect is used (see [2]).

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

boolean – If f is None, whether self is full;
otherwise, whether self is full of $$f$$-palindromes.

EXAMPLES:

sage: words.ThueMorseWord()[:100].is_full()
False
sage: words.FibonacciWord()[:100].is_full()
True
sage: Word('000000000000000').is_full()
True
sage: Word('011010011001').is_full()
False
sage: Word('2194').is_full()
True
sage: Word().is_full()
True

sage: f = WordMorphism('a->b,b->a')
sage: Word().is_full(f)
True
sage: w = Word('ab')
sage: w.is_full()
True
sage: w.is_full(f)
True

sage: f = WordMorphism('a->b,b->a')
sage: Word('abab').is_full(f)
True
sage: Word('abba').is_full(f)
False


A simple example of an infinite word full of f-palindromes:

sage: p = WordMorphism({0:'abc',1:'ab'})
sage: f = WordMorphism('a->b,b->a,c->c')
sage: p(words.FibonacciWord()[:50]).is_full(f)
True
sage: p(words.FibonacciWord()[:150]).is_full(f)
True


REFERENCES:

• [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
• [2] E. Pelantová, Š. Starosta, Infinite words rich and almost rich in generalized palindromes, in: G. Mauri, A. Leporati (Eds.), Developments in Language Theory, volume 6795 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 406–416
is_smooth_prefix()

Returns True if self is the prefix of a smooth word, and False otherwise.

Let $$A_k = \{1, \ldots ,k\}$$, $$k \geq 2$$. An infinite word $$w$$ in $$A_k^\omega$$ is said to be smooth if and only if for all positive integers $$m$$, $$\Delta^m(w)$$ is in $$A_k^\omega$$, where $$\Delta(w)$$ is the word obtained from $$w$$ by composing the length of consecutive runs of the same letter in $$w$$. See for instance [1] and [2].

INPUT:

• self - must be a word over the integers to get something other than False

OUTPUT:

boolean – whether self is a smooth prefix or not

EXAMPLES:

sage: W = Words([1, 2])
sage: W([1, 1, 2, 2, 1, 2, 1, 1]).is_smooth_prefix()
True
sage: W([1, 2, 1, 2, 1, 2]).is_smooth_prefix()
False


REFERENCES:

• [1] S. Brlek, A. Ladouceur, A note on differentiable palindromes, Theoret. Comput. Sci. 302 (2003) 167–178.
• [2] S. Brlek, S. Dulucq, A. Ladouceur, L. Vuillon, Combinatorial properties of smooth infinite words, Theoret. Comput. Sci. 352 (2006) 306–317.
is_square()

Returns True if self is a square, and False otherwise.

EXAMPLES:

sage: Word([1,0,0,1]).is_square()
False
sage: Word('1212').is_square()
True
sage: Word('1213').is_square()
False
sage: Word('12123').is_square()
False
sage: Word().is_square()
True

is_square_free()

Returns True if self does not contain squares, and False otherwise.

EXAMPLES:

sage: Word('12312').is_square_free()
True
sage: Word('31212').is_square_free()
False
sage: Word().is_square_free()
True


TESTS:

We make sure that #8490 is fixed:

sage: Word('11').is_square_free()
False
sage: Word('211').is_square_free()
False
sage: Word('3211').is_square_free()
False

is_sturmian_factor()

Tells whether self is a factor of a Sturmian word.

The finite word self must be defined on a two-letter alphabet.

Equivalently, tells whether self is balanced. The advantage over the is_balanced method is that this one runs in linear time whereas is_balanced runs in quadratic time.

OUTPUT:

• boolean – the result.

EXAMPLES:

sage: w = Word('0111011011011101101',alphabet='01')
sage: w.is_sturmian_factor()
True

sage: words.LowerMechanicalWord(random(),alphabet='01')[:100].is_sturmian_factor()
True
sage: words.CharacteristicSturmianWord(random())[:100].is_sturmian_factor()
True

sage: w = Word('aabb',alphabet='ab')
sage: w.is_sturmian_factor()
False

sage: s1 = WordMorphism('a->ab,b->b')
sage: s2 = WordMorphism('a->ba,b->b')
sage: s3 = WordMorphism('a->a,b->ba')
sage: s4 = WordMorphism('a->a,b->ab')
sage: W = Words('ab')
sage: w = W('ab')
sage: for i in xrange(8): w = choice([s1,s2,s3,s4])(w)
sage: w
word: abaaabaaabaabaaabaaabaabaaabaabaaabaaaba...
sage: w.is_sturmian_factor()
True


Famous words:

sage: words.FibonacciWord()[:100].is_sturmian_factor()
True
sage: words.ThueMorseWord()[:1000].is_sturmian_factor()
False
sage: words.KolakoskiWord()[:1000].is_sturmian_factor()
False


REFERENCES:

 [Arn2002] P. Arnoux, Sturmian sequences, in Substitutions in Dynamics, N. Pytheas Fogg (Ed.), Arithmetics, and Combinatorics (Lecture Notes in Mathematics, Vol. 1794), 2002.
 [Ser1985] C. Series. The geometry of Markoff numbers. The Mathematical Intelligencer, 7(3):20–29, 1985.
 [SU2009] J. Smillie and C. Ulcigrai. Symbolic coding for linear trajectories in the regular octagon, Arxiv 0905.0871, 2009.

AUTHOR:

• Thierry Monteil
is_subword_of(other)

Returns True is self is a subword of other, and False otherwise.

EXAMPLES:

sage: Word().is_subword_of(Word('123'))
True
sage: Word('123').is_subword_of(Word('3211333213233321'))
True
sage: Word('321').is_subword_of(Word('11122212112122133111222332'))
False

is_suffix(other)

Returns True if w is a suffix of other, and False otherwise.

EXAMPLES:

sage: w = Word('0123456789')
sage: y = Word('56789')
sage: y.is_suffix(w)
True
sage: w.is_suffix(y)
False
sage: Word('579').is_suffix(w)
False
sage: Word().is_suffix(y)
True
sage: w.is_suffix(Word())
False
sage: Word().is_suffix(Word())
True

is_suffix_of(other)

Returns True if w is a suffix of other, and False otherwise.

EXAMPLES:

sage: w = Word('0123456789')
sage: y = Word('56789')
sage: y.is_suffix_of(w)
doctest:1: DeprecationWarning: is_suffix_of is deprecated, use is_suffix instead!
See http://trac.sagemath.org/8429 for details.
True
sage: w.is_suffix_of(y)
False
sage: Word('579').is_suffix_of(w)
False
sage: Word().is_suffix_of(y)
True
sage: w.is_suffix_of(Word())
False
sage: Word().is_suffix_of(Word())
True

is_symmetric(f=None)

Returns True if self is symmetric (or $$f$$-symmetric), and False otherwise.

A word is symmetric (resp. $$f$$-symmetric) if it is the product of two palindromes (resp. $$f$$-palindromes). See [1] and [2].

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

EXAMPLES:

sage: Word('abbabab').is_symmetric()
True
sage: Word('ababa').is_symmetric()
True
sage: Word('aababaabba').is_symmetric()
False
sage: Word('aabbbaababba').is_symmetric()
False
sage: f = WordMorphism('a->b,b->a')
sage: Word('aabbbaababba').is_symmetric(f)
True


REFERENCES:

• [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
• [2] A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
is_tangent()

Tells whether self is a tangent word.

The finite word self must be defined on a two-letter alphabet.

A binary word is said to be tangent if it can appear in infintely many cutting sequences of a smooth curve, where each cutting sequence is observed on a progressively smaller grid.

This class of words strictly contains the class of 1-balanced words, and is strictly contained in the class of 2-balanced words.

This method runs in linear time.

OUTPUT:

• boolean – the result.

EXAMPLES:

sage: w = Word('01110110110111011101',alphabet='01')
sage: w.is_tangent()
True


Some tangent words may not be balanced:

sage: Word('aabb',alphabet='ab').is_balanced()
False
sage: Word('aabb',alphabet='ab').is_tangent()
True


Some 2-balanced words may not be tangent:

sage: Word('aaabb',alphabet='ab').is_tangent()
False
sage: Word('aaabb',alphabet='ab').is_balanced(2)
True


Famous words:

sage: words.FibonacciWord()[:100].is_tangent()
True
sage: words.ThueMorseWord()[:1000].is_tangent()
True
sage: words.KolakoskiWord()[:1000].is_tangent()
False


REFERENCES:

 [Mon2010] T. Monteil, The asymptotic language of smooth curves, talk at LaCIM2010.

AUTHOR:

• Thierry Monteil
iterated_left_palindromic_closure(f=None)

Returns the iterated left ($$f$$-)palindromic closure of self.

INPUT:

• f – involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

word – the left iterated $$f$$-palindromic closure of self.

EXAMPLES:

sage: Word('123').iterated_left_palindromic_closure()
word: 3231323
sage: f = WordMorphism('a->b,b->a')
sage: Word('ab').iterated_left_palindromic_closure(f=f)
word: abbaab
sage: Word('aab').iterated_left_palindromic_closure(f=f)
word: abbaabbaab


TESTS:

If f is not a involution:

sage: f = WordMorphism('a->b,b->b')
sage: Word('aab').iterated_left_palindromic_closure(f=f)
Traceback (most recent call last):
...
TypeError: self (=a->b, b->b) is not an endomorphism


REFERENCES:

• A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
iterated_palindromic_closure(side='right', f=None)

Returns the iterated ($$f$$-)palindromic closure of self.

INPUT:

• side'right' or 'left' (default: 'right') the direction of the closure
• f – involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

word – If $$f$$ is None, the right iterated palindromic closure of self; otherwise, the right iterated $$f$$-palindromic closure of self. If side is 'left', the left palindromic closure.

EXAMPLES:

sage: Word('123').iterated_palindromic_closure()
doctest:1: DeprecationWarning: iterated_palindromic_closure is deprecated, use iterated_left_palindromic_closure or iterated_right_palindromic_closure instead!
See http://trac.sagemath.org/8429 for details.
word: 1213121
sage: Word('123').iterated_palindromic_closure(side='left')
word: 3231323
sage: Word('1').iterated_palindromic_closure()
word: 1
sage: Word().iterated_palindromic_closure()
word:
sage: Word = Words('ab')
sage: f = WordMorphism('a->b,b->a')
sage: Word('ab').iterated_palindromic_closure(f=f)
word: abbaab
sage: Word('ab').iterated_palindromic_closure(f=f, side='left')
word: abbaab
sage: Word('aab').iterated_palindromic_closure(f=f)
word: ababbaabab
sage: Word('aab').iterated_palindromic_closure(f=f, side='left')
word: abbaabbaab


TESTS:

sage: Word('aab').iterated_palindromic_closure(f=f, side='leftt')
Traceback (most recent call last):
...
ValueError: side must be either 'left' or 'right' (not leftt)

If f is not an involution:
sage: f = WordMorphism(‘a->b,b->b’) sage: Word(‘aab’).iterated_palindromic_closure(f=f, side=’left’) Traceback (most recent call last): ... TypeError: self (=a->b, b->b) is not an endomorphism

REFERENCES:

• A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
lacunas(f=None)

Returns the list of all the lacunas of self.

A lacuna is a position in a word where the longest ($$f$$-)palindromic suffix is not unioccurrent (see [1]).

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e., $$f$$ equal to the identity.

OUTPUT:

list – list of all the lacunas of self.

EXAMPLES:

sage: w = Word([0,1,1,2,3,4,5,1,13,3])
sage: w.lacunas()
[7, 9]
sage: words.ThueMorseWord()[:100].lacunas()
[8, 9, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 96, 97, 98, 99]
sage: f = WordMorphism({0:[1],1:[0]})
sage: words.ThueMorseWord()[:50].lacunas(f)
[0, 2, 4, 12, 16, 17, 18, 19, 48, 49]


REFERENCES:

• [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena, Arezzo-Italia), 53–67.
last_position_dict()

Returns a dictionary that contains the last position of each letter in self.

EXAMPLES:

sage: Word('1231232').last_position_dict()
{'1': 3, '3': 5, '2': 6}

last_position_table()

Returns a dictionary that contains the last position of each letter in self.

EXAMPLES:

sage: Word('1231232').last_position_table()
doctest:1: DeprecationWarning: last_position_table is deprecated, use last_position_dict instead!
See http://trac.sagemath.org/8429 for details.
{'1': 3, '3': 5, '2': 6}

left_special_factors(n=None)

Returns the left special factors (of length n).

A factor $$u$$ of a word $$w$$ is left special if there are two distinct letters $$a$$ and $$b$$ such that $$au$$ and $$bu$$ are factors of $$w$$.

INPUT:

• n - integer (optional, default: None). If None, it returns all left special factors.

OUTPUT:

A list of words.

EXAMPLES:

sage: alpha, beta, x = 0.54, 0.294, 0.1415
sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40]
sage: for i in range(5): print i, sorted(w.left_special_factors(i))
0 [word: ]
1 [word: 0]
2 [word: 00, word: 01]
3 [word: 000, word: 010]
4 [word: 0000, word: 0101]

left_special_factors_iterator(n=None)

Returns an iterator over the left special factors (of length n).

A factor $$u$$ of a word $$w$$ is left special if there are two distinct letters $$a$$ and $$b$$ such that $$au$$ and $$bu$$ are factors of $$w$$.

INPUT:

• n - integer (optional, default: None). If None, it returns an iterator over all left special factors.

EXAMPLES:

sage: alpha, beta, x = 0.54, 0.294, 0.1415
sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40]
sage: sorted(w.left_special_factors_iterator(3))
[word: 000, word: 010]
sage: sorted(w.left_special_factors_iterator(4))
[word: 0000, word: 0101]
sage: sorted(w.left_special_factors_iterator(5))
[word: 00000, word: 01010]

length()

Returns the length of self.

TESTS:

sage: from sage.combinat.words.word import Word_class
sage: w = Word(iter('abba'*40), length="finite")
sage: w._len is None
True
sage: w.length()
160
sage: w = Word(iter('abba'), length=4)
sage: w._len
4
sage: w.length()
4
sage: def f(n):
...     return range(2,12,2)[n]
sage: w = Word(f, length=5)
sage: w.length()
5

length_border()

Returns the length of the border of self.

The border of a word is the longest word that is both a proper prefix and a proper suffix of self.

EXAMPLES:

sage: Word('121').length_border()
1
sage: Word('1').length_border()
0
sage: Word('1212').length_border()
2
sage: Word('111').length_border()
2
sage: Word().length_border() is None
True

lengths_lps(f=None)

Returns the list of the length of the longest palindromic suffix (lps) for each non-empty prefix of self.

It corresponds to the function $$G_w$$ defined in [1].

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

list – list of the length of the longest palindromic
suffix (lps) for each non-empty prefix of self.

EXAMPLES:

sage: Word().lengths_lps()
[]
sage: Word('a').lengths_lps()
[1]
sage: Word('aaa').lengths_lps()
[1, 2, 3]
sage: Word('abbabaabbaab').lengths_lps()
[1, 1, 2, 4, 3, 3, 2, 4, 2, 4, 6, 8]

sage: f = WordMorphism('a->b,b->a')
sage: Word('abbabaabbaab').lengths_lps(f)
[0, 2, 0, 2, 2, 4, 6, 8, 4, 6, 4, 6]

sage: f = WordMorphism({5:[8],8:[5]})
sage: Word([5,8,5,5,8,8,5,5,8,8,5,8,5]).lengths_lps(f)
[0, 2, 2, 0, 2, 4, 6, 4, 6, 8, 10, 12, 4]


REFERENCES:

• [1] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi, Reconstructing words from a fixed palindromic length sequence, Proc. TCS 2008, 5th IFIP International Conference on Theoretical Computer Science (September 8-10 2008, Milano, Italia), accepted.
lengths_unioccurrent_lps(f=None)

Returns the list of the lengths of the unioccurrent longest ($$f$$)-palindromic suffixes (lps) for each non-empty prefix of self. No unioccurrent lps are indicated by None.

It corresponds to the function $$H_w$$ defined in [1] and [2].

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e., $$f$$ equal to the identity.

OUTPUT:

list – list of the length of the unioccurrent longest palindromic
suffix (lps) for each non-empty prefix of self. No unioccurrent lps are indicated by None.

EXAMPLES:

sage: w = Word([0,1,1,2,3,4,5,1,13,3])
sage: w.lengths_unioccurrent_lps()
[1, 1, 2, 1, 1, 1, 1, None, 1, None]
sage: f = words.FibonacciWord()[:20]
sage: f.lengths_unioccurrent_lps() == f.lengths_lps()
True
sage: t = words.ThueMorseWord()
sage: t[:20].lengths_unioccurrent_lps()
[1, 1, 2, 4, 3, 3, 2, 4, None, None, 6, 8, 10, 12, 14, 16, 6, 8, 10, 12]
sage: f = WordMorphism({1:[0],0:[1]})
sage: t[:15].lengths_unioccurrent_lps(f)
[None, 2, None, 2, None, 4, 6, 8, 4, 6, 4, 6, None, 4, 6]


REFERENCES:

• [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena, Arezzo-Italia), 53–67.
• [2] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi, Reconstructing words from a fixed palindromic length sequence, Proc. TCS 2008, 5th IFIP International Conference on Theoretical Computer Science (September 8-10 2008, Milano, Italia), accepted.
letters()

Return a list of the letters that appear in self, listed in the order of first appearance.

EXAMPLES:

sage: Word([0,1,1,0,1,0,0,1]).letters()
[0, 1]
sage: Word("cacao").letters()
['c', 'a', 'o']

longest_common_subword(other)

Returns a longest subword of self and other.

A subword of a word is a subset of the word’s letters, read in the order in which they appear in the word.

INPUT:

• other – a word

ALGORITHM:

For any indices $$i,j$$, we compute the longest common subword lcs[i,j] of $$self[:i]$$ and $$other[:j]$$. This can be easily obtained as the longest of

• lcs[i-1,j]
• lcs[i,j-1]
• lcs[i-1,j-1]+self[i] if self[i]==other[j]

EXAMPLES:

sage: v1 = Word("abc")
sage: v2 = Word("ace")
sage: v1.longest_common_subword(v2)
word: ac

sage: w1 = Word("1010101010101010101010101010101010101010")
sage: w2 = Word("0011001100110011001100110011001100110011")
sage: w1.longest_common_subword(w2)
word: 00110011001100110011010101010


TESTS:

sage: Word().longest_common_subword(Word())
word:


is_subword_of()

longest_common_suffix(other)

Returns the longest common suffix of self and other.

EXAMPLES:

sage: w = Word('112345678')
sage: u = Word('1115678')
sage: w.longest_common_suffix(u)
word: 5678
sage: u.longest_common_suffix(u)
word: 1115678
sage: u.longest_common_suffix(w)
word: 5678
sage: w.longest_common_suffix(w)
word: 112345678
sage: y = Word('549332345')
sage: w.longest_common_suffix(y)
word:


TESTS:

With the empty word:

sage: w.longest_common_suffix(Word())
word:
sage: Word().longest_common_suffix(w)
word:
sage: Word().longest_common_suffix(Word())
word:


With an infinite word:

sage: t=words.ThueMorseWord('ab')
sage: w.longest_common_suffix(t)
Traceback (most recent call last):
...
TypeError: other must be a finite word

lps(f=None, l=None)

Returns the longest palindromic (or $$f$$-palindromic) suffix of self.

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
• l - integer (default: None) the length of the longest palindrome suffix of self[:-1], if known.

OUTPUT:

word – If f is None, the longest palindromic suffix of self;
otherwise, the longest f-palindromic suffix of self.

EXAMPLES:

sage: Word('0111').lps()
word: 111
sage: Word('011101').lps()
word: 101
sage: Word('6667').lps()
word: 7
sage: Word('abbabaab').lps()
word: baab
sage: Word().lps()
word:
sage: f = WordMorphism('a->b,b->a')
sage: Word('abbabaab').lps(f=f)
word: abbabaab
sage: w = Word('33412321')
sage: w.lps(l=3)
word: 12321
sage: Y = Word
sage: w = Y('01101001')
sage: w.lps(l=2)
word: 1001
sage: w.lps()
word: 1001
sage: w.lps(l=None)
word: 1001
sage: Y().lps(l=2)
Traceback (most recent call last):
...
IndexError: list index out of range
sage: v = Word('abbabaab')
sage: pal = v[:0]
sage: for i in range(1, v.length()+1):
...     pal = v[:i].lps(l=pal.length())
...     pal
...
word: a
word: b
word: bb
word: abba
word: bab
word: aba
word: aa
word: baab
sage: f = WordMorphism('a->b,b->a')
sage: v = Word('abbabaab')
sage: pal = v[:0]
sage: for i in range(1, v.length()+1):
...     pal = v[:i].lps(f=f, l=pal.length())
...     pal
...
word:
word: ab
word:
word: ba
word: ab
word: baba
word: bbabaa
word: abbabaab

lyndon_factorization()

Returns the Lyndon factorization of self.

The Lyndon factorization of a finite word $$w$$ is the unique factorization of $$w$$ as a non-increasing product of Lyndon words, i.e., $$w = l_1\cdots l_n$$ where each $$l_i$$ is a Lyndon word and $$l_1\geq \cdots \geq l_n$$. See for instance [1].

OUTPUT:

list – the list of factors obtained

EXAMPLES:

sage: Word('010010010001000').lyndon_factorization()
(01, 001, 001, 0001, 0, 0, 0)
sage: Words('10')('010010010001000').lyndon_factorization()
(0, 10010010001000)
sage: Word('abbababbaababba').lyndon_factorization()
(abb, ababb, aababb, a)
sage: Words('ba')('abbababbaababba').lyndon_factorization()
(a, bbababbaaba, bba)
sage: Word([1,2,1,3,1,2,1]).lyndon_factorization()
(1213, 12, 1)


TESTS:

sage: Words('01')('').lyndon_factorization()
()
sage: Word('01').lyndon_factorization()
(01)
sage: Words('10')('01').lyndon_factorization()
(0, 1)
sage: lynfac = Word('abbababbaababba').lyndon_factorization()
sage: [x.is_lyndon() for x in lynfac]
[True, True, True, True]
sage: lynfac = Words('ba')('abbababbaababba').lyndon_factorization()
sage: [x.is_lyndon() for x in lynfac]
[True, True, True]
sage: w = words.ThueMorseWord()[:1000]
sage: w == prod(w.lyndon_factorization())
True


REFERENCES:

• [1] J.-P. Duval, Factorizing words over an ordered alphabet, J. Algorithms 4 (1983) 363–381.
• [2] G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42.
minimal_period()

Returns the period of self.

Let $$A$$ be an alphabet. An integer $$p\geq 1$$ is a period of a word $$w=a_1a_2\cdots a_n$$ where $$a_i\in A$$ if $$a_i=a_{i+p}$$ for $$i=1,\ldots,n-p$$. The smallest period of $$w$$ is called the period of $$w$$. See Chapter 1 of [1].

EXAMPLES:

sage: Word('aba').minimal_period()
2
sage: Word('abab').minimal_period()
2
sage: Word('ababa').minimal_period()
2
sage: Word('ababaa').minimal_period()
5
sage: Word('ababac').minimal_period()
6
sage: Word('aaaaaa').minimal_period()
1
sage: Word('a').minimal_period()
1
sage: Word().minimal_period()
1


REFERENCES:

• [1] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
nb_factor_occurrences_in(other)

Returns the number of times self appears as a factor in other.

EXAMPLES:

sage: Word().nb_factor_occurrences_in(Word('123'))
Traceback (most recent call last):
...
NotImplementedError: The factor must be non empty
sage: Word('123').nb_factor_occurrences_in(Word('112332312313112332121123'))
4
sage: Word('321').nb_factor_occurrences_in(Word('11233231231311233221123'))
0

nb_subword_occurrences_in(other)

Returns the number of times self appears in other as a subword.

EXAMPLES:

sage: Word().nb_subword_occurrences_in(Word('123'))
Traceback (most recent call last):
...
NotImplementedError: undefined value
sage: Word('123').nb_subword_occurrences_in(Word('1133432311132311112'))
11
sage: Word('4321').nb_subword_occurrences_in(Word('1132231112233212342231112'))
0
sage: Word('3').nb_subword_occurrences_in(Word('122332112321213'))
4

number_of_factors(n=None)

Counts the number of distinct factors of self.

INPUT:

• n - an integer, or None.

OUTPUT:

If n is an integer, returns the number of distinct factors of length n. If n is None, returns the total number of distinct factors.

EXAMPLES:

sage: w = Word([1,2,1,2,3])
sage: w.number_of_factors()
13
sage: map(w.number_of_factors, range(6))
[1, 3, 3, 3, 2, 1]

sage: w = words.ThueMorseWord()[:100]
sage: [w.number_of_factors(i) for i in range(10)]
[1, 2, 4, 6, 10, 12, 16, 20, 22, 24]

sage: Word('1213121').number_of_factors()
22
sage: Word('1213121').number_of_factors(1)
3

sage: Word('a'*100).number_of_factors()
101
sage: Word('a'*100).number_of_factors(77)
1

sage: Word().number_of_factors()
1
sage: Word().number_of_factors(17)
0

sage: blueberry = Word("blueberry")
sage: blueberry.number_of_factors()
43
sage: map(blueberry.number_of_factors, range(10))
[1, 6, 8, 7, 6, 5, 4, 3, 2, 1]

number_of_left_special_factors(n)

Returns the number of left special factors of length n.

A factor $$u$$ of a word $$w$$ is left special if there are two distinct letters $$a$$ and $$b$$ such that $$au$$ and $$bu$$ are factors of $$w$$.

INPUT:

• n - integer

OUTPUT:

Non negative integer

EXAMPLES:

sage: w = words.FibonacciWord()[:100]
sage: [w.number_of_left_special_factors(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

sage: w = words.ThueMorseWord()[:100]
sage: [w.number_of_left_special_factors(i) for i in range(10)]
[1, 2, 2, 4, 2, 4, 4, 2, 2, 4]

number_of_right_special_factors(n)

Returns the number of right special factors of length n.

A factor $$u$$ of a word $$w$$ is right special if there are two distinct letters $$a$$ and $$b$$ such that $$ua$$ and $$ub$$ are factors of $$w$$.

INPUT:

• n - integer

OUTPUT:

Non negative integer

EXAMPLES:

sage: w = words.FibonacciWord()[:100]
sage: [w.number_of_right_special_factors(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

sage: w = words.ThueMorseWord()[:100]
sage: [w.number_of_right_special_factors(i) for i in range(10)]
[1, 2, 2, 4, 2, 4, 4, 2, 2, 4]

order()

Returns the order of self.

Let $$p(w)$$ be the period of a word $$w$$. The positive rational number $$|w|/p(w)$$ is the order of $$w$$. See Chapter 8 of [1].

OUTPUT:

rational – the order

EXAMPLES:

sage: Word('abaaba').order()
2
sage: Word('ababaaba').order()
8/5
sage: Word('a').order()
1
sage: Word('aa').order()
2
sage: Word().order()
0


REFERENCES:

• [1] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
overlap_partition(other, delay=0, p=None, involution=None)

Returns the partition of the alphabet induced by the overlap of self and other with the given delay.

The partition of the alphabet is given by the equivalence relation obtained from the symmetric, reflexive and transitive closure of the set of pairs of letters $$R_{u,v,d} = \{ (u_k, v_{k-d}) : 0 \leq k < n, 0\leq k-d < m \}$$ where $$u = u_0 u_1 \cdots u_{n-1}$$, $$v = v_0v_1\cdots v_{m-1}$$ are two words on the alphabet $$A$$ and $$d$$ is an integer.

The equivalence relation defined by $$R$$ is inspired from [1].

INPUT:

• other - word on the same alphabet as self
• delay - integer
• p - disjoint sets data structure (optional, default: None), a partition of the alphabet into disjoint sets to start with. If None, each letter start in distinct equivalence classes.
• involution - callable (optional, default: None), an involution on the alphabet. If involution is not None, the relation $$R_{u,v,d} \cup R_{involution(u),involution(v),d}$$ is considered.

OUTPUT:

• disjoint set data structure

EXAMPLES:

sage: W = Words(list('abc')+range(6))
sage: u = W('abc')
sage: v = W(range(5))
sage: u.overlap_partition(v)
{{0, 'a'}, {1, 'b'}, {2, 'c'}, {3}, {4}, {5}}
sage: u.overlap_partition(v, 2)
{{'a'}, {'b'}, {0, 'c'}, {1}, {2}, {3}, {4}, {5}}
sage: u.overlap_partition(v, -1)
{{0}, {1, 'a'}, {2, 'b'}, {3, 'c'}, {4}, {5}}


You can re-use the same disjoint set and do more than one overlap:

sage: p = u.overlap_partition(v, 2)
sage: p
{{'a'}, {'b'}, {0, 'c'}, {1}, {2}, {3}, {4}, {5}}
sage: u.overlap_partition(v, 1, p)
{{'a'}, {0, 1, 'b', 'c'}, {2}, {3}, {4}, {5}}


The function overlap_partition can be used to study equations on words. For example, if a word $$w$$ overlaps itself with delay $$d$$, then $$d$$ is a period of $$w$$:

sage: W = Words(range(20))
sage: w = W(range(14)); w
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13
sage: d = 5
sage: p = w.overlap_partition(w, d)
sage: m = WordMorphism(p.element_to_root_dict())
sage: w2 = m(w); w2
word: 56789567895678
sage: w2.minimal_period() == d
True


If a word is equal to its reversal, then it is a palindrome:

sage: W = Words(range(20))
sage: w = W(range(17)); w
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
sage: p = w.overlap_partition(w.reversal(), 0)
sage: m = WordMorphism(p.element_to_root_dict())
sage: w2 = m(w); w2
word: 01234567876543210
sage: w2.parent()
Words over {0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 18, 19}
sage: w2.is_palindrome()
True


If the reversal of a word $$w$$ is factor of its square $$w^2$$, then $$w$$ is symmetric, i.e. the product of two palindromes:

sage: W = Words(range(10))
sage: w = W(range(10)); w
word: 0123456789
sage: p = (w*w).overlap_partition(w.reversal(), 4)
sage: m = WordMorphism(p.element_to_root_dict())
sage: w2 = m(w); w2
word: 0110456654
sage: w2.is_symmetric()
True


If the image of the reversal of a word $$w$$ under an involution $$f$$ is factor of its square $$w^2$$, then $$w$$ is $$f$$-symmetric:

sage: W = Words([-11,-9,..,11])
sage: w = W([1,3,..,11])
sage: w
word: 1,3,5,7,9,11
sage: inv = lambda x:-x
sage: f = WordMorphism(dict( (a, inv(a)) for a in W.alphabet()))
sage: p = (w*w).overlap_partition(f(w).reversal(), 2, involution=f)
sage: m = WordMorphism(p.element_to_root_dict())
sage: m(w)
word: 1,-1,5,7,-7,-5
sage: m(w).is_symmetric(f)
True


TESTS:

sage: W = Words('abcdef')
sage: w = W('abc')
sage: y = W('def')
sage: w.overlap_partition(y, -3)
{{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}
sage: w.overlap_partition(y, -2)
{{'a', 'f'}, {'b'}, {'c'}, {'d'}, {'e'}}
sage: w.overlap_partition(y, -1)
{{'a', 'e'}, {'b', 'f'}, {'c'}, {'d'}}
sage: w.overlap_partition(y, 0)
{{'a', 'd'}, {'b', 'e'}, {'c', 'f'}}
sage: w.overlap_partition(y, 1)
{{'a'}, {'b', 'd'}, {'c', 'e'}, {'f'}}
sage: w.overlap_partition(y, 2)
{{'a'}, {'b'}, {'c', 'd'}, {'e'}, {'f'}}
sage: w.overlap_partition(y, 3)
{{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}
sage: w.overlap_partition(y, 4)
{{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}

sage: W = Words(range(2))
sage: w = W([0,1,0,1,0,1]); w
word: 010101
sage: w.overlap_partition(w, 0)
{{0}, {1}}
sage: w.overlap_partition(w, 1)
{{0, 1}}

sage: empty = Word()
sage: empty.overlap_partition(empty, 'yo')
Traceback (most recent call last):
...
TypeError: delay (=yo) must be an integer
sage: empty.overlap_partition(empty,2,'yo')
Traceback (most recent call last):
...
TypeError: p(=yo) is not a DisjointSet


The involution input can be any callable:

sage: w = Words([-5,..,5])([-5..5])
sage: inv = lambda x:-x
sage: w.overlap_partition(w, 2, involution=inv)
{{-4, -2, 0, 2, 4}, {-5, -3, -1, 1, 3, 5}}


REFERENCES:

• [1] S. Labbé, Propriétés combinatoires des $$f$$-palindromes, Mémoire de maîtrise en Mathématiques, Montréal, UQAM, 2008, 109 pages.
palindrome_prefixes()

Returns a list of all palindrome prefixes of self.

OUTPUT:

list – A list of all palindrome prefixes of self.

EXAMPLES:

sage: w = Word('abaaba')
sage: w.palindrome_prefixes()
[word: , word: a, word: aba, word: abaaba]
sage: w = Word('abbbbbbbbbb')
sage: w.palindrome_prefixes()
[word: , word: a]

palindromes(f=None)

Returns the set of all palindromic (or $$f$$-palindromic) factors of self.

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

set – If f is None, the set of all palindromic factors of self;
otherwise, the set of all f-palindromic factors of self.

EXAMPLES:

sage: sorted(Word('01101001').palindromes())
[word: , word: 0, word: 00, word: 010, word: 0110, word: 1, word: 1001, word: 101, word: 11]
sage: sorted(Word('00000').palindromes())
[word: , word: 0, word: 00, word: 000, word: 0000, word: 00000]
sage: sorted(Word('0').palindromes())
[word: , word: 0]
sage: sorted(Word('').palindromes())
[word: ]
sage: sorted(Word().palindromes())
[word: ]
sage: f = WordMorphism('a->b,b->a')
sage: sorted(Word('abbabaab').palindromes(f))
[word: , word: ab, word: abbabaab, word: ba, word: baba, word: bbabaa]

palindromic_closure(side='right', f=None)

Return the shortest palindrome having self as a prefix (or as a suffix if side is 'left').

See [1].

INPUT:

• side'right' or 'left' (default: 'right') the direction of the closure
• f – involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

OUTPUT:

word – If f is None, the right palindromic closure of self;
otherwise, the right f-palindromic closure of self. If side is 'left', the left palindromic closure.

EXAMPLES:

sage: Word('1233').palindromic_closure()
word: 123321
sage: Word('12332').palindromic_closure()
word: 123321
sage: Word('0110343').palindromic_closure()
word: 01103430110
sage: Word('0110343').palindromic_closure(side='left')
word: 3430110343
sage: Word('01105678').palindromic_closure(side='left')
word: 876501105678
sage: w = Word('abbaba')
sage: w.palindromic_closure()
word: abbababba

sage: f = WordMorphism('a->b,b->a')
sage: w.palindromic_closure(f=f)
word: abbabaab
sage: w.palindromic_closure(f=f, side='left')
word: babaabbaba


TESTS:

sage: f = WordMorphism('a->c,c->a')
sage: w.palindromic_closure(f=f, side='left')
Traceback (most recent call last):
...
KeyError: 'b'


REFERENCES:

• [1] A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
palindromic_lacunas_study(f=None)

Returns interesting statistics about longest ($$f$$-)palindromic suffixes and lacunas of self (see [1] and [2]).

Note that a word $$w$$ has at most $$|w| + 1$$ different palindromic factors (see [3]). For $$f$$-palindromes (or pseudopalidromes or theta-palindromes), the maximum number of $$f$$-palindromic factors is $$|w|+1-g_f(w)$$, where $$g_f(w)$$ is the number of pairs $$\{a, f(a)\}$$ such that $$a$$ is a letter, $$a$$ is not equal to $$f(a)$$, and $$a$$ or $$f(a)$$ occurs in $$w$$, see [4].

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e., $$f$$ equal to the identity.

OUTPUT:

• list - list of the length of the longest palindromic suffix (lps) for each non-empty prefix of self;
• list - list of all the lacunas, i.e. positions where there is no unioccurrent lps;
• set - set of palindromic factors of self.

EXAMPLES:

sage: a,b,c = Word('abbabaabbaab').palindromic_lacunas_study()
sage: a
[1, 1, 2, 4, 3, 3, 2, 4, 2, 4, 6, 8]
sage: b
[8, 9]
sage: c          # random order
set([word: , word: b, word: bab, word: abba, word: bb, word: aa, word: baabbaab, word: baab, word: aba, word: aabbaa, word: a])

sage: f = WordMorphism('a->b,b->a')
sage: a,b,c = Word('abbabaab').palindromic_lacunas_study(f=f)
sage: a
[0, 2, 0, 2, 2, 4, 6, 8]
sage: b
[0, 2, 4]
sage: c           # random order
set([word: , word: ba, word: baba, word: ab, word: bbabaa, word: abbabaab])
sage: c == set([Word(), Word('ba'), Word('baba'), Word('ab'), Word('bbabaa'), Word('abbabaab')])
True


REFERENCES:

• [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena, Arezzo-Italia), 53–67.
• [2] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi, Reconstructing words from a fixed palindromic length sequence, Proc. TCS 2008, 5th IFIP International Conference on Theoretical Computer Science (September 8-10 2008, Milano, Italia), accepted.
• [3] X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001) 539–553.
• [4] Š. Starosta, On Theta-palindromic Richness, Theoret. Comp. Sci. 412 (2011) 1111–1121
parikh_vector(alphabet=None)

Returns the Parikh vector of self, i.e., the vector containing the number of occurrences of each letter, given in the order of the alphabet.

INPUT:

• alphabet - (default: None) finite ordered alphabet, if None it uses the set of letters in self with the ordering defined by the parent

EXAMPLES:

sage: Words('ab')().parikh_vector()
[0, 0]
sage: Word('aabaa').parikh_vector('abc')
[4, 1, 0]
sage: Word('a').parikh_vector('abc')
[1, 0, 0]
sage: Word('a').parikh_vector('cab')
[0, 1, 0]
sage: Word('a').parikh_vector('bca')
[0, 0, 1]
sage: Word().parikh_vector('ab')
[0, 0]
sage: Word().parikh_vector('abc')
[0, 0, 0]
sage: Word().parikh_vector('abcd')
[0, 0, 0, 0]


TESTS:

sage: Word('aabaa').parikh_vector()
Traceback (most recent call last):
...
TypeError: the alphabet is infinite; specify a finite alphabet or use evaluation_dict() instead

periods(divide_length=False)

Returns a list containing the periods of self between $$1$$ and $$n - 1$$, where $$n$$ is the length of self.

INPUT:

• divide_length - boolean (default: False). When set to True, then only periods that divide the length of self are considered.

OUTPUT:

List of positive integers

EXAMPLES:

sage: w = Word('ababab')
sage: w.periods()
[2, 4]
sage: w.periods(divide_length=True)
[2]
sage: w = Word('ababa')
sage: w.periods()
[2, 4]
sage: w.periods(divide_length=True)
[]

phi()

Applies the phi function to self and returns the result. This is the word obtained by taking the first letter of the words obtained by iterating delta on self.

OUTPUT:

word – the result of the phi function

EXAMPLES:

sage: W = Words([1, 2])
sage: W([2,2,1,1,2,1,2,2,1,2,2,1,1,2]).phi()
word: 222222
sage: W([2,1,2,2,1,2,2,1,2,1]).phi()
word: 212113
sage: W().phi()
word:
sage: Word([2,1,2,2,1,2,2,1,2,1]).phi()
word: 212113
sage: Word([2,3,1,1,2,1,2,3,1,2,2,3,1,2]).phi()
word: 21215
sage: Word("aabbabaabaabba").phi()
word: a22222
sage: w = Word([2,3,1,1,2,1,2,3,1,2,2,3,1,2])


REFERENCES:

• S. Brlek, A. Ladouceur, A note on differentiable palindromes, Theoret. Comput. Sci. 302 (2003) 167–178.
• S. Brlek, S. Dulucq, A. Ladouceur, L. Vuillon, Combinatorial properties of smooth infinite words, Theoret. Comput. Sci. 352 (2006) 306–317.
phi_inv(W=None)

Apply the inverse of the phi function to self.

INPUT:

• self - a word over the integers
• W - a parent object of words defined over integers

OUTPUT:

word – the inverse of the phi function

EXAMPLES:

sage: W = Words([1, 2])
sage: W([2, 2, 2, 2, 1, 2]).phi_inv()
word: 22112122
sage: W([2, 2, 2]).phi_inv(Words([2, 3]))
word: 2233

prefix_function_table()

Returns a vector containing the length of the proper prefix-suffixes for all the non-empty prefixes of self.

EXAMPLES:

sage: Word('121321').prefix_function_table()
[0, 0, 1, 0, 0, 1]
sage: Word('1241245').prefix_function_table()
[0, 0, 0, 1, 2, 3, 0]
sage: Word().prefix_function_table()
[]

primitive()

Returns the primitive of self.

EXAMPLES:

sage: Word('12312').primitive()
word: 12312
sage: Word('121212').primitive()
word: 12

primitive_length()

Returns the length of the primitive of self.

EXAMPLES:

sage: Word('1231').primitive_length()
4
sage: Word('121212').primitive_length()
2

quasiperiods()

Returns the quasiperiods of self as a list ordered from shortest to longest.

Let $$w$$ be a finite or infinite word. A quasiperiod of $$w$$ is a proper factor $$u$$ of $$w$$ such that the occurrences of $$u$$ in $$w$$ entirely cover $$w$$, i.e., every position of $$w$$ falls within some occurrence of $$u$$ in $$w$$. See for instance [1], [2], and [3].

EXAMPLES:

sage: Word('abaababaabaababaaba').quasiperiods()
[word: aba, word: abaaba, word: abaababaaba]
sage: Word('abaaba').quasiperiods()
[word: aba]
sage: Word('abacaba').quasiperiods()
[]


REFERENCES:

• [1] A. Apostolico, A. Ehrenfeucht, Efficient detection of quasiperiodicities in strings, Theoret. Comput. Sci. 119 (1993) 247–265.
• [2] S. Marcus, Quasiperiodic infinite words, Bull. Eur. Assoc. Theor. Comput. Sci. 82 (2004) 170-174.
• [3] A. Glen, F. Levé, G. Richomme, Quasiperiodic and Lyndon episturmian words, Preprint, 2008, arXiv:0805.0730.
rauzy_graph(n)

Returns the Rauzy graph of the factors of length n of self.

The vertices are the factors of length $$n$$ and there is an edge from $$u$$ to $$v$$ if $$ua = bv$$ is a factor of length $$n+1$$ for some letters $$a$$ and $$b$$.

INPUT:

• n - integer

EXAMPLES:

sage: w = Word(range(10)); w
word: 0123456789
sage: g = w.rauzy_graph(3); g
Looped digraph on 8 vertices
sage: WordOptions(identifier='')
sage: g.vertices()
[012, 123, 234, 345, 456, 567, 678, 789]
sage: g.edges()
[(012, 123, 3),
(123, 234, 4),
(234, 345, 5),
(345, 456, 6),
(456, 567, 7),
(567, 678, 8),
(678, 789, 9)]
sage: WordOptions(identifier='word: ')

sage: f = words.FibonacciWord()[:100]
sage: f.rauzy_graph(8)
Looped digraph on 9 vertices

sage: w = Word('1111111')
sage: g = w.rauzy_graph(3)
sage: g.edges()
[(word: 111, word: 111, word: 1)]

sage: w = Word('111')
sage: for i in range(5) : w.rauzy_graph(i)
Looped multi-digraph on 1 vertex
Looped digraph on 1 vertex
Looped digraph on 1 vertex
Looped digraph on 1 vertex
Looped digraph on 0 vertices


Multi-edges are allowed for the empty word:

sage: W = Words('abcde')
sage: w = W('abc')
sage: w.rauzy_graph(0)
Looped multi-digraph on 1 vertex
sage: _.edges()
[(word: , word: , word: a),
(word: , word: , word: b),
(word: , word: , word: c)]

reduced_rauzy_graph(n)

Returns the reduced Rauzy graph of order $$n$$ of self.

INPUT:

• n - non negative integer. Every vertex of a reduced Rauzy graph of order $$n$$ is a factor of length $$n$$ of self.

OUTPUT:

Looped multi-digraph

DEFINITION:

For infinite periodic words (resp. for finite words of type $$u^i u[0:j]$$), the reduced Rauzy graph of order $$n$$ (resp. for $$n$$ smaller or equal to $$(i-1)|u|+j$$) is the directed graph whose unique vertex is the prefix $$p$$ of length $$n$$ of self and which has an only edge which is a loop on $$p$$ labelled by $$w[n+1:|w|] p$$ where $$w$$ is the unique return word to $$p$$.

In other cases, it is the directed graph defined as followed. Let $$G_n$$ be the Rauzy graph of order $$n$$ of self. The vertices are the vertices of $$G_n$$ that are either special or not prolongable to the right or to the left. For each couple ($$u$$, $$v$$) of such vertices and each directed path in $$G_n$$ from $$u$$ to $$v$$ that contains no other vertices that are special, there is an edge from $$u$$ to $$v$$ in the reduced Rauzy graph of order $$n$$ whose label is the label of the path in $$G_n$$.

Note

In the case of infinite recurrent non periodic words, this definition correspond to the following one that can be found in [1] and [2] where a simple path is a path that begins with a special factor, ends with a special factor and contains no other vertices that are special:

The reduced Rauzy graph of factors of length $$n$$ is obtained from $$G_n$$ by replacing each simple path $$P=v_1 v_2 ... v_{\ell}$$ with an edge $$v_1 v_{\ell}$$ whose label is the concatenation of the labels of the edges of $$P$$.

EXAMPLES:

sage: w = Word(range(10)); w
word: 0123456789
sage: g = w.reduced_rauzy_graph(3); g
Looped multi-digraph on 2 vertices
sage: g.vertices()
[word: 012, word: 789]
sage: g.edges()
[(word: 012, word: 789, word: 3456789)]


For the Fibonacci word:

sage: f = words.FibonacciWord()[:100]
sage: g = f.reduced_rauzy_graph(8);g
Looped multi-digraph on 2 vertices
sage: g.vertices()
[word: 01001010, word: 01010010]
sage: g.edges()
[(word: 01001010, word: 01010010, word: 010), (word: 01010010, word: 01001010, word: 01010), (word: 01010010, word: 01001010, word: 10)]


For periodic words:

sage: from itertools import cycle
sage: w = Word(cycle('abcd'))[:100]
sage: g = w.reduced_rauzy_graph(3)
sage: g.edges()
[(word: abc, word: abc, word: dabc)]

sage: w = Word('111')
sage: for i in range(5) : w.reduced_rauzy_graph(i)
Looped digraph on 1 vertex
Looped digraph on 1 vertex
Looped digraph on 1 vertex
Looped multi-digraph on 1 vertex
Looped multi-digraph on 0 vertices


For ultimately periodic words:

sage: sigma = WordMorphism('a->abcd,b->cd,c->cd,d->cd')
sage: w = sigma.fixed_point('a')[:100]; w
word: abcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd...
sage: g = w.reduced_rauzy_graph(5)
sage: g.vertices()
[word: abcdc, word: cdcdc]
sage: g.edges()
[(word: abcdc, word: cdcdc, word: dc), (word: cdcdc, word: cdcdc, word: dc)]


AUTHOR:

Julien Leroy (March 2010): initial version

REFERENCES:

• [1] M. Bucci et al. A. De Luca, A. Glen, L. Q. Zamboni, A connection between palindromic and factor complexity using return words,” Advances in Applied Mathematics 42 (2009) 60-74.
• [2] L’ubomira Balkova, Edita Pelantova, and Wolfgang Steiner. Sequences with constant number of return words. Monatsh. Math, 155 (2008) 251-263.
return_words(fact)

Returns the set of return words of fact in self.

This is the set of all factors starting by the given factor and ending just before the next occurrence of this factor. See [1] and [2].

INPUT:

• fact - a non empty finite word

OUTPUT:

Python set of finite words

EXAMPLES:

sage: Word('21331233213231').return_words(Word('2'))
set([word: 213, word: 21331, word: 233])
sage: Word().return_words(Word('213'))
set([])
sage: Word('121212').return_words(Word('1212'))
set([word: 12])

sage: TM = words.ThueMorseWord()[:10000]
sage: TM.return_words(Word([0]))     # optional long time (1.34 s)
set([word: 0, word: 01, word: 011])


REFERENCES:

• [1] F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998) 89-101.
• [2] C. Holton, L.Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Syst. 32 (1999) 133-157.
return_words_derivate(fact)

Returns the word generated by mapping a letter to each occurrence of the return words for the given factor dropping any dangling prefix and suffix. See for instance [1].

EXAMPLES:

sage: Word('12131221312313122').return_words_derivate(Word('1'))
word: 123242


REFERENCES:

• [1] F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998) 89–101.
rev_lex_less(other)

Returns True if the word self is reverse lexicographically less than other.

EXAMPLES:

sage: Word([1,2,4]).rev_lex_less(Word([1,3,2]))
True
sage: Word([3,2,1]).rev_lex_less(Word([1,2,3]))
False

reversal()

Returns the reversal of self.

EXAMPLES:

sage: Word('124563').reversal()
word: 365421

rfind(sub, start=0, end=None)

Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure.

INPUT:

• sub - string or word to search for.
• start - non negative integer (default: 0) specifying the position at which the search must stop.
• end - non negative integer (default: None) specifying the position from which to start the search. If None, then the search is performed up to the end of the string.

OUTPUT:

non negative integer or -1

EXAMPLES:

sage: w = Word([0,1,0,0,1])
sage: w.rfind(Word([0,1]))
3
sage: w.rfind(Word([0,1]), end=4)
0
sage: w.rfind(Word([0,1]), end=5)
3
sage: w.rfind(Word([0,0]), start=2, end=5)
2
sage: w.rfind(Word([0,0]), start=3, end=5) == -1
True


Instances of Word_str handle string inputs as well:

sage: w = Word('abac')
sage: w.rfind('a')
2
sage: w.rfind(Word('a'))
2

right_special_factors(n=None)

Returns the right special factors (of length n).

A factor $$u$$ of a word $$w$$ is right special if there are two distinct letters $$a$$ and $$b$$ such that $$ua$$ and $$ub$$ are factors of $$w$$.

INPUT:

• n - integer (optional, default: None). If None, it returns all right special factors.

OUTPUT:

A list of words.

EXAMPLES:

sage: w = words.ThueMorseWord()[:30]
sage: for i in range(5): print i, sorted(w.right_special_factors(i))
0 [word: ]
1 [word: 0, word: 1]
2 [word: 01, word: 10]
3 [word: 001, word: 010, word: 101, word: 110]
4 [word: 0110, word: 1001]

right_special_factors_iterator(n=None)

Returns an iterator over the right special factors (of length n).

A factor $$u$$ of a word $$w$$ is right special if there are two distinct letters $$a$$ and $$b$$ such that $$ua$$ and $$ub$$ are factors of $$w$$.

INPUT:

• n - integer (optional, default: None). If None, it returns an iterator over all right special factors.

EXAMPLES:

sage: alpha, beta, x = 0.61, 0.54, 0.3
sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40]
sage: sorted(w.right_special_factors_iterator(3))
[word: 010, word: 101]
sage: sorted(w.right_special_factors_iterator(4))
[word: 0101, word: 1010]
sage: sorted(w.right_special_factors_iterator(5))
[word: 00101, word: 11010]

robinson_schensted()

Return the semistandard tableau and standard tableau pair obtained by running the Robinson-Schensted algorithm on self.

This can also be done by running RSK() on self.

EXAMPLES:

sage: Word([1,1,3,1,2,3,1]).robinson_schensted()
[[[1, 1, 1, 1, 3], [2], [3]], [[1, 2, 3, 5, 6], [4], [7]]]

schuetzenberger_involution(n=None)
Returns the Schuetzenberger involution of the word self, which is obtained by reverting the word and then complementing all letters within the underlying ordered alphabet. If $$n$$ is specified, the underlying alphabet is assumed to be $$[1,2,\ldots,n]$$. If no alphabet is specified, $$n$$ is the maximal letter appearing in self.

INPUT:

• self – a word
• n – an integer specifying the maximal letter in the alphabet (optional)

OUTPUT:

• a word, the Schuetzenberger involution of self

EXAMPLES:

sage: w = Word([9,7,4,1,6,2,3])
sage: v = w.schuetzenberger_involution(); v
word: 7849631
sage: v.parent()
Words

sage: w = Word([1,2,3],alphabet=[1,2,3,4,5])
sage: v = w.schuetzenberger_involution();v
word: 345
sage: v.parent()
Words over {1, 2, 3, 4, 5}

sage: w = Word([1,2,3])
sage: v = w.schuetzenberger_involution(n=5);v
word: 345
sage: v.parent()
Words

sage: w = Word([11,32,69,2,53,1,2,3,18,41])
sage: w.schuetzenberger_involution()
word: 29,52,67,68,69,17,68,1,38,59

sage: w = Word([],alphabet=[1,2,3,4,5])
sage: w.schuetzenberger_involution()
word:

sage: w = Word([])
sage: w.schuetzenberger_involution()
word:

shifted_shuffle(other, shift=None)

Returns the combinatorial class representing the shifted shuffle product between words self and other. This is the same as the shuffle product of self with the word obtained from other by incrementing its values (i.e. its letters) by the given shift.

INPUT:

• other - finite word over the integers
• shift - integer or None (default: None) added to each letter of other. When shift is None, it is replaced by self.length()

OUTPUT:

Combinatorial class of shifted shuffle products of self and other.

EXAMPLES:

sage: w = Word([0,1,1])
sage: sp = w.shifted_shuffle(w); sp
Shuffle product of word: 011 and word: 344
sage: sp = w.shifted_shuffle(w, 2); sp
Shuffle product of word: 011 and word: 233
sage: sp.cardinality()
20
sage: WordOptions(identifier='')
sage: sp.list()
[011233, 012133, 012313, 012331, 021133, 021313, 021331, 023113, 023131, 023311, 201133, 201313, 201331, 203113, 203131, 203311, 230113, 230131, 230311, 233011]
sage: WordOptions(identifier='word: ')
sage: y = Word('aba')
sage: y.shifted_shuffle(w,2)
Traceback (most recent call last):
...
ValueError: for shifted shuffle, words must only contain integers as letters

shuffle(other, overlap=0)

Returns the combinatorial class representing the shuffle product between words self and other. This consists of all words of length self.length()+other.length() that have both self and other as subwords.

If overlap is non-zero, then the combinatorial class representing the shuffle product with overlaps is returned. The calculation of the shift in each overlap is done relative to the order of the alphabet. For example, “a” shifted by “a” is “b” in the alphabet [a, b, c] and 0 shifted by 1 in [0, 1, 2, 3] is 2.

INPUT:

• other - finite word
• overlap - (default: 0) integer or True

OUTPUT:

Combinatorial class of shuffle product of self and other

EXAMPLES:

sage: ab = Word("ab")
sage: cd = Word("cd")
sage: sp = ab.shuffle(cd); sp
Shuffle product of word: ab and word: cd
sage: sp.cardinality()
6
sage: sp.list()
[word: abcd, word: acbd, word: acdb, word: cabd, word: cadb, word: cdab]
sage: w = Word([0,1])
sage: u = Word([2,3])
sage: w.shuffle(w)
Shuffle product of word: 01 and word: 01
sage: u.shuffle(u)
Shuffle product of word: 23 and word: 23
sage: w.shuffle(u)
Shuffle product of word: 01 and word: 23
sage: w.shuffle(u,2)
Overlapping shuffle product of word: 01 and word: 23 with 2 overlaps

size_of_alphabet()

EXAMPLES:

sage: w = Word('abaccefa')
sage: w.size_of_alphabet()
doctest:1: DeprecationWarning: size_of_alphabet() is deprecated, use parent().size_of_alphabet() instead!
See http://trac.sagemath.org/8429 for details.
+Infinity
sage: y = Words('456')('64654564')
sage: y.size_of_alphabet()
3

standard_factorization()

Returns the standard factorization of self.

The standard factorization of a word $$w$$ of length greater than 1 is the unique factorization: $$w = uv$$ where $$v$$ is the longest proper suffix of $$w$$ that is a Lyndon word.

Note that if $$w$$ is a Lyndon word of length greater than 1 with standard factorization $$w = uv$$, then $$u$$ and $$v$$ are also Lyndon words and $$u < v$$.

See for instance [1], [2] and [3].

INPUT:

• self - finite word of length greater than 1

OUTPUT:

tuple – tuple of two factors

EXAMPLES:

sage: Words('01')('0010110011').standard_factorization()
(word: 001011, word: 0011)
sage: Words('123')('1223312').standard_factorization()
(word: 12233, word: 12)
sage: Word([3,2,1]).standard_factorization()
(word: 32, word: 1)

sage: w = Word('0010110011',alphabet='01')
sage: w.standard_factorization()
(word: 001011, word: 0011)
sage: w = Word('0010110011',alphabet='10')
sage: w.standard_factorization()
(word: 001011001, word: 1)
sage: w = Word('1223312',alphabet='123')
sage: w.standard_factorization()
(word: 12233, word: 12)


TESTS:

sage: w = Word()
sage: w.standard_factorization()
Traceback (most recent call last):
...
ValueError: Standard factorization not defined on words of
length less than 2
sage: w = Word('a')
sage: w.standard_factorization()
Traceback (most recent call last):
...
ValueError: Standard factorization not defined on words of
length less than 2


REFERENCES:

• [1] K.-T. Chen, R.H. Fox, R.C. Lyndon, Free differential calculus, IV. The quotient groups of the lower central series, Ann. of Math. 68 (1958) 81–95.
• [2] J.-P. Duval, Factorizing words over an ordered alphabet, J. Algorithms 4 (1983) 363–381.
• [3] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
standard_factorization_of_lyndon_factorization()

Returns the standard factorization of the Lyndon factorization of self.

OUTPUT:

list of lists – the factorization

EXAMPLES:

sage: Words('123')('1221131122').standard_factorization_of_lyndon_factorization()
doctest:...: DeprecationWarning: standard_factorization_of_lyndon_factorization(): is deprecated since March 2012 and will be removed in a later version of Sage.
See http://trac.sagemath.org/12469 for details.
[(word: 12, word: 2), (word: 1, word: 13), (word: 1, word: 122)]

standard_permutation()

Returns the standard permutation of the word self on the ordered alphabet. It is defined as the permutation with exactly the same number of inversions as w. Equivalently, it is the permutation of minimal length whose inverse sorts self.

EXAMPLES:

sage: w = Word([1,2,3,2,2,1]); w
word: 123221
sage: p = w.standard_permutation(); p
[1, 3, 6, 4, 5, 2]
sage: v = Word(p.inverse().action(w)); v
word: 112223
sage: filter(lambda q: q.length() <= p.length() and \
....:       q.inverse().action(w) == list(v), \
....:       Permutations(w.length()) )
[[1, 3, 6, 4, 5, 2]]

sage: w = Words([1,2,3])([1,2,3,2,2,1,2,1]); w
word: 12322121
sage: p = w.standard_permutation(); p
[1, 4, 8, 5, 6, 2, 7, 3]
sage: Word(p.inverse().action(w))
word: 11122223

sage: w = Words([3,2,1])([1,2,3,2,2,1,2,1]); w
word: 12322121
sage: p = w.standard_permutation(); p
[6, 2, 1, 3, 4, 7, 5, 8]
sage: Word(p.inverse().action(w))
word: 32222111

sage: w = Words('ab')('abbaba'); w
word: abbaba
sage: p = w.standard_permutation(); p
[1, 4, 5, 2, 6, 3]
sage: Word(p.inverse().action(w))
word: aaabbb

sage: w = Words('ba')('abbaba'); w
word: abbaba
sage: p = w.standard_permutation(); p
[4, 1, 2, 5, 3, 6]
sage: Word(p.inverse().action(w))
word: bbbaaa

sturmian_desubstitute_as_possible()

Sturmian desubstitutes the word self as much as possible.

The finite word self must be defined on a two-letter alphabet or use at most two-letters.

It can be Sturmian desubstituted if one letter appears isolated: the Sturmian desubstitution consists in removing one letter per run of the non-isolated letter. The accelerated Sturmian desubstitution consists in removing a run equal to the length of the shortest inner run from any run of the non-isolated letter (including possible leading and trailing runs even if they have shorter length). The (accelerated) Sturmian desubstitution is done as much as possible. A word is a factor of a Sturmian word if, and only if, the result is the empty word.

OUTPUT:

• A finite word defined on a two-letter alphabet.

EXAMPLES:

sage: u = Word('10111101101110111',alphabet='01') ; u
word: 10111101101110111
sage: v = u.sturmian_desubstitute_as_possible() ; v
word: 01100101
sage: v == v.sturmian_desubstitute_as_possible()
True

sage: Word('azaazaaazaaazaazaaaz', alphabet='az').sturmian_desubstitute_as_possible()
word:


TESTS:

sage: w = Word('azazaza', alphabet='aze')
sage: w.sturmian_desubstitute_as_possible()
word:
sage: Word('aze').sturmian_desubstitute_as_possible()
Traceback (most recent call last):
...
TypeError: your word must be defined on a binary alphabet or use at most two different letters
sage: Word('azaaazaazaazaaazaaza', alphabet='az').sturmian_desubstitute_as_possible()
word:
sage: Word('azaaazaazaazaaazaaaza', alphabet='az').sturmian_desubstitute_as_possible()
word: azzaa


Boundary effects:

sage: Word('', alphabet='az').sturmian_desubstitute_as_possible()
word:
sage: Word('azzzzz', alphabet='az').sturmian_desubstitute_as_possible()
word:
sage: Word('zzzzza', alphabet='az').sturmian_desubstitute_as_possible()
word:
sage: Word('aaaaazaaaaaaaaa', alphabet='az').sturmian_desubstitute_as_possible()
word:
sage: Word('aaaaaaaaaaaaaa', alphabet='az').sturmian_desubstitute_as_possible()
word:


Boundary effects without alphabet:

sage: Word('').sturmian_desubstitute_as_possible()
word:
sage: Word('azzzzz').sturmian_desubstitute_as_possible()
word:
sage: Word('zzzzza').sturmian_desubstitute_as_possible()
word:
sage: Word('aaaaazaaaaaaaaa').sturmian_desubstitute_as_possible()
word:
sage: Word('aaaaaaaaaaaaaa').sturmian_desubstitute_as_possible()
word:


Idempotence:

sage: r = words.RandomWord(randint(1,15)).sturmian_desubstitute_as_possible() ; r == r.sturmian_desubstitute_as_possible()
True


AUTHOR:

• Thierry Monteil
suffix_tree()

Alias for implicit_suffix_tree().

EXAMPLES:

sage: Word('abbabaab').suffix_tree()
Implicit Suffix Tree of the word: abbabaab

suffix_trie()

Returns the suffix trie of self.

The suffix trie of a finite word $$w$$ is a data structure representing the factors of $$w$$. It is a tree whose edges are labelled with letters of $$w$$, and whose leafs correspond to suffixes of $$w$$.

EXAMPLES:

sage: w = Word("cacao")
sage: w.suffix_trie()
Suffix Trie of the word: cacao

sage: w = Word([0,1,0,1,1])
sage: w.suffix_trie()
Suffix Trie of the word: 01011

swap(i, j=None)

Returns the word w with entries at positions i and j swapped. By default, j = i+1.

EXAMPLES:

sage: Word([1,2,3]).swap(0,2)
word: 321
sage: Word([1,2,3]).swap(1)
word: 132
sage: Word("abba").swap(1,-1)
word: aabb

swap_decrease(i)

Returns the word with positions i and i+1 exchanged if self[i] < self[i+1]. Otherwise, it returns self.

EXAMPLES:

sage: w = Word([1,3,2])
sage: w.swap_decrease(0)
word: 312
sage: w.swap_decrease(1)
word: 132
sage: w.swap_decrease(1) is w
True
sage: Words("ab")("abba").swap_decrease(0)
word: baba
sage: Words("ba")("abba").swap_decrease(0)
word: abba

swap_increase(i)

Returns the word with positions i and i+1 exchanged if self[i] > self[i+1]. Otherwise, it returns self.

EXAMPLES:

sage: w = Word([1,3,2])
sage: w.swap_increase(1)
word: 123
sage: w.swap_increase(0)
word: 132
sage: w.swap_increase(0) is w
True
sage: Words("ab")("abba").swap_increase(0)
word: abba
sage: Words("ba")("abba").swap_increase(0)
word: baba

to_integer_list()

Returns a list of integers from [0,1,...,self.length()-1] in the same relative order as the letters in self in the parent.

EXAMPLES:

sage: from itertools import count
sage: w = Word('abbabaab')
sage: w.to_integer_list()
[0, 1, 1, 0, 1, 0, 0, 1]
sage: w = Word(iter("cacao"), length="finite")
sage: w.to_integer_list()
[1, 0, 1, 0, 2]
sage: w = Words([3,2,1])([2,3,3,1])
sage: w.to_integer_list()
[1, 0, 0, 2]

to_integer_word()

Returns a word defined over the integers [0,1,...,self.length()-1] whose letters are in the same relative order in the parent.

EXAMPLES:

sage: from itertools import count
sage: w = Word('abbabaab')
sage: w.to_integer_word()
word: 01101001
sage: w = Word(iter("cacao"), length="finite")
sage: w.to_integer_word()
word: 10102
sage: w = Words([3,2,1])([2,3,3,1])
sage: w.to_integer_word()
word: 1002

to_monoid_element()

Return self as an element the free monoid with the same alphabet as self.

EXAMPLES:

sage: w = Word('aabb')
sage: w.to_monoid_element()
a^2*b^2
sage: W = Words('abc')
sage: w = W(w)
sage: w.to_monoid_element()
a^2*b^2


TESTS:

Check that w == w.to_monoid_element().to_word():

sage: all(w.to_monoid_element().to_word() == w for i in range(6) for w in Words('abc', i))
True

topological_entropy(n)

Return the topological entropy for the factors of length n.

The topological entropy of a sequence $$u$$ is defined as the exponential growth rate of the complexity of $$u$$ as the length increases: $$H_{top}(u)=\lim_{n\to\infty}\frac{\log_d(p_u(n))}{n}$$ where $$d$$ denotes the cardinality of the alphabet and $$p_u(n)$$ is the complexity function, i.e. the number of factors of length $$n$$ in the sequence $$u$$ [1].

INPUT:

• self - a word defined over a finite alphabet
• n - positive integer

OUTPUT:

real number (a symbolic expression)

EXAMPLES:

sage: W = Words([0, 1])
sage: w = W([0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1])
sage: t = w.topological_entropy(3); t
1/3*log(7)/log(2)
sage: n(t)
0.935784974019201

sage: w = words.ThueMorseWord()[:100]
sage: topo = w.topological_entropy
sage: for i in range(0, 41, 5): print i, n(topo(i), digits=5)
0 1.0000
5 0.71699
10 0.48074
15 0.36396
20 0.28774
25 0.23628
30 0.20075
35 0.17270
40 0.14827


If no alphabet is specified, an error is raised:

sage: w = Word(range(20))
sage: w.topological_entropy(3)
Traceback (most recent call last):
...
TypeError: The word must be defined over a finite alphabet


The following is ok:

sage: W = Words(range(20))
sage: w = W(range(20))
sage: w.topological_entropy(3)
1/3*log(18)/log(20)


REFERENCES:

[1] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics, and Combinatorics, Lecture Notes in Mathematics 1794, Springer Verlag. V. Berthe, S. Ferenczi, C. Mauduit and A. Siegel, Eds. (2002).