(Non-negative) Integer vectors

AUTHORS:

  • Mike Hanson (2007) - original module
  • Nathann Cohen, David Joyner (2009-2010) - Gale-Ryser stuff
  • Nathann Cohen, David Joyner (2011) - Gale-Ryser bugfix
  • Travis Scrimshaw (2012-05-12) - Updated doc-strings to tell the user of that the class’s name is a misnomer (that they only contains non-negative entries).
  • Federico Poloni (2013) - specialized rank()
sage.combinat.integer_vector.IntegerVectors(n=None, k=None, **kwargs)

Returns the combinatorial class of (non-negative) integer vectors.

NOTE - These integer vectors are non-negative.

EXAMPLES: If n is not specified, it returns the class of all integer vectors.

sage: IntegerVectors()
Integer vectors
sage: [] in IntegerVectors()
True
sage: [1,2,1] in IntegerVectors()
True
sage: [1, 0, 0] in IntegerVectors()
True

Entries are non-negative.

sage: [-1, 2] in IntegerVectors()
False

If n is specified, then it returns the class of all integer vectors which sum to n.

sage: IV3 = IntegerVectors(3); IV3
Integer vectors that sum to 3

Note that trailing zeros are ignored so that [3, 0] does not show up in the following list (since [3] does)

sage: IntegerVectors(3, max_length=2).list()
[[3], [2, 1], [1, 2], [0, 3]]

If n and k are both specified, then it returns the class of integer vectors that sum to n and are of length k.

sage: IV53 = IntegerVectors(5,3); IV53
Integer vectors of length 3 that sum to 5
sage: IV53.cardinality()
21
sage: IV53.first()
[5, 0, 0]
sage: IV53.last()
[0, 0, 5]
sage: IV53.random_element()
[4, 0, 1]
class sage.combinat.integer_vector.IntegerVectors_all(category=None, *keys, **opts)

Bases: sage.combinat.combinat.CombinatorialClass

TESTS:

sage: C = sage.combinat.combinat.CombinatorialClass()
sage: C.category()
Category of enumerated sets
sage: C.__class__
<class 'sage.combinat.combinat.CombinatorialClass_with_category'>
sage: isinstance(C, Parent)
True
sage: C = sage.combinat.combinat.CombinatorialClass(category = FiniteEnumeratedSets())
sage: C.category()
Category of finite enumerated sets
cardinality()

EXAMPLES:

sage: IntegerVectors().cardinality()
+Infinity
list()

EXAMPLES:

sage: IntegerVectors().list()
Traceback (most recent call last):
...
NotImplementedError: infinite list
class sage.combinat.integer_vector.IntegerVectors_nconstraints(n, constraints)

Bases: sage.combinat.integer_vector.IntegerVectors_nkconstraints

TESTS:

sage: IV = IntegerVectors(3, max_length=2)
sage: IV == loads(dumps(IV))
True
cardinality()

EXAMPLES:

sage: IntegerVectors(3, max_length=2).cardinality()
4
sage: IntegerVectors(3).cardinality()
+Infinity
list()

EXAMPLES:

sage: IntegerVectors(3, max_length=2).list()
[[3], [2, 1], [1, 2], [0, 3]]
sage: IntegerVectors(3).list()
Traceback (most recent call last):
...
NotImplementedError: infinite list
class sage.combinat.integer_vector.IntegerVectors_nk(n, k)

Bases: sage.combinat.combinat.CombinatorialClass

TESTS:

sage: IV = IntegerVectors(2,3)
sage: IV == loads(dumps(IV))
True

AUTHORS:

  • Martin Albrecht
  • Mike Hansen
list()

EXAMPLE:

sage: IV = IntegerVectors(2,3)
sage: IV.list()
[[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]
sage: IntegerVectors(3, 0).list()
[]
sage: IntegerVectors(3, 1).list()
[[3]]
sage: IntegerVectors(0, 1).list()
[[0]]
sage: IntegerVectors(0, 2).list()
[[0, 0]]
sage: IntegerVectors(2, 2).list()
[[2, 0], [1, 1], [0, 2]]
rank(x)

Returns the position of a given element.

INPUT:

  • x - a list with sum(x) == n and len(x) == k

TESTS:

sage: IV = IntegerVectors(4,5) 
sage: range(IV.cardinality()) == [IV.rank(x) for x in IV]
True
class sage.combinat.integer_vector.IntegerVectors_nkconstraints(n, k, constraints)

Bases: sage.combinat.combinat.CombinatorialClass

EXAMPLES:

sage: IV = IntegerVectors(2,3,min_slope=0)
sage: IV == loads(dumps(IV))
True
cardinality()

EXAMPLES:

sage: IntegerVectors(3,3, min_part=1).cardinality()
1
sage: IntegerVectors(5,3, min_part=1).cardinality()
6
sage: IntegerVectors(13, 4, min_part=2, max_part=4).cardinality()
16
first()

EXAMPLES:

sage: IntegerVectors(2,3,min_slope=0).first()
[0, 1, 1]
next(x)

EXAMPLES:

sage: IntegerVectors(2,3,min_slope=0).last()
[0, 0, 2]
class sage.combinat.integer_vector.IntegerVectors_nnondescents(n, comp)

Bases: sage.combinat.combinat.CombinatorialClass

The combinatorial class of integer vectors v graded by two parameters:

  • n: the sum of the parts of v
  • comp: the non descents composition of v

In other words: the length of v equals c[1]+...+c[k], and v is decreasing in the consecutive blocs of length c[1], ..., c[k]

Those are the integer vectors of sum n which are lexicographically maximal (for the natural left->right reading) in their orbit by the young subgroup S_c_1 x x S_c_k. In particular, they form a set of orbit representative of integer vectors w.r.t. this young subgroup.

sage.combinat.integer_vector.constant_func(i)

Returns the constant function i.

EXAMPLES:

sage: f = sage.combinat.integer_vector.constant_func(3)
sage: f(-1)
3
sage: f('asf')
3
sage.combinat.integer_vector.gale_ryser_theorem(p1, p2, algorithm='gale')

Returns the binary matrix given by the Gale-Ryser theorem.

The Gale Ryser theorem asserts that if \(p_1,p_2\) are two partitions of \(n\) of respective lengths \(k_1,k_2\), then there is a binary \(k_1\times k_2\) matrix \(M\) such that \(p_1\) is the vector of row sums and \(p_2\) is the vector of column sums of \(M\), if and only if the conjugate of \(p_2\) dominates \(p_1\).

INPUT:

  • p1, p2– list of integers representing the vectors of row/column sums

  • algorithm – two possible string values :

    • "ryser" implements the construction due to Ryser [Ryser63].
    • "gale" (default) implements the construction due to Gale [Gale57].

OUTPUT:

  • A binary matrix if it exists, None otherwise.

Gale’s Algorithm:

(Gale [Gale57]): A matrix satisfying the constraints of its sums can be defined as the solution of the following Linear Program, which Sage knows how to solve.

\[\begin{split}\forall i&\sum_{j=1}^{k_2} b_{i,j}=p_{1,j}\\ \forall i&\sum_{j=1}^{k_1} b_{j,i}=p_{2,j}\\ &b_{i,j}\mbox{ is a binary variable}\end{split}\]

Ryser’s Algorithm:

(Ryser [Ryser63]): The construction of an \(m\times n\) matrix \(A=A_{r,s}\), due to Ryser, is described as follows. The construction works if and only if have \(s\preceq r^*\).

  • Construct the \(m\times n\) matrix \(B\) from \(r\) by defining the \(i\)-th row of \(B\) to be the vector whose first \(r_i\) entries are \(1\), and the remainder are 0’s, \(1\leq i\leq m\). This maximal matrix \(B\) with row sum \(r\) and ones left justified has column sum \(r^{*}\).
  • Shift the last \(1\) in certain rows of \(B\) to column \(n\) in order to achieve the sum \(s_n\). Call this \(B\) again.
    • The \(1\)‘s in column n are to appear in those rows in which \(A\) has the largest row sums, giving preference to the bottom-most positions in case of ties.
    • Note: When this step automatically “fixes” other columns, one must skip ahead to the first column index with a wrong sum in the step below.
  • Proceed inductively to construct columns \(n-1\), ..., \(2\), \(1\).
  • Set \(A = B\). Return \(A\).

EXAMPLES:

Computing the matrix for \(p_1=p_2=2+2+1\)

sage: from sage.combinat.integer_vector import gale_ryser_theorem
sage: p1 = [2,2,1]
sage: p2 = [2,2,1]
sage: print gale_ryser_theorem(p1, p2)     # not tested
[1 1 0]
[1 0 1]
[0 1 0]
sage: A = gale_ryser_theorem(p1, p2)
sage: rs = [sum(x) for x in A.rows()]
sage: cs = [sum(x) for x in A.columns()]
sage: p1 == rs; p2 == cs
True
True

Or for a non-square matrix with \(p_1=3+3+2+1\) and \(p_2=3+2+2+1+1\), using Ryser’s algorithm

sage: from sage.combinat.integer_vector import gale_ryser_theorem
sage: p1 = [3,3,1,1]
sage: p2 = [3,3,1,1]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 0 1]
[1 1 1 0]
[0 1 0 0]
[1 0 0 0]
sage: p1 = [4,2,2]
sage: p2 = [3,3,1,1]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 1 1]
[1 1 0 0]
[1 1 0 0]
sage: p1 = [4,2,2,0]
sage: p2 = [3,3,1,1,0,0]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 1 1 0 0]
[1 1 0 0 0 0]
[1 1 0 0 0 0]
[0 0 0 0 0 0]
sage: p1 = [3,3,2,1]
sage: p2 = [3,2,2,1,1]
sage: print gale_ryser_theorem(p1, p2, algorithm="gale")  # not tested
[1 1 1 0 0]
[1 1 0 0 1]
[1 0 1 0 0]
[0 0 0 1 0]

With \(0\) in the sequences, and with unordered inputs

sage: from sage.combinat.integer_vector import gale_ryser_theorem
sage: gale_ryser_theorem([3,3,0,1,1,0], [3,1,3,1,0], algorithm = "ryser")
[1 0 1 1 0]
[1 1 1 0 0]
[0 0 0 0 0]
[0 0 1 0 0]
[1 0 0 0 0]
[0 0 0 0 0]
sage: p1 = [3,1,1,1,1]; p2 = [3,2,2,0]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 1 0]
[0 0 1 0]
[0 1 0 0]
[1 0 0 0]
[1 0 0 0]

TESTS:

This test created a random bipartite graph on \(n+m\) vertices. Its adjacency matrix is binary, and it is used to create some “random-looking” sequences which correspond to an existing matrix. The gale_ryser_theorem is then called on these sequences, and the output checked for correctness.:

sage: def test_algorithm(algorithm, low = 10, high = 50):
...      n,m = randint(low,high), randint(low,high)
...      g = graphs.RandomBipartite(n, m, .3)
...      s1 = sorted(g.degree([(0,i) for i in range(n)]), reverse = True)
...      s2 = sorted(g.degree([(1,i) for i in range(m)]), reverse = True)
...      m = gale_ryser_theorem(s1, s2, algorithm = algorithm)
...      ss1 = sorted(map(lambda x : sum(x) , m.rows()), reverse = True)
...      ss2 = sorted(map(lambda x : sum(x) , m.columns()), reverse = True)
...      if ((ss1 != s1) or (ss2 != s2)):
...          print "Algorithm %s failed with this input:" % algorithm
...          print s1, s2

sage: for algorithm in ["gale", "ryser"]:                            # long time
...      for i in range(50):                                         # long time
...         test_algorithm(algorithm, 3, 10)                         # long time

Null matrix:

sage: gale_ryser_theorem([0,0,0],[0,0,0,0], algorithm="gale")
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
sage: gale_ryser_theorem([0,0,0],[0,0,0,0], algorithm="ryser")
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]

REFERENCES:

[Ryser63](1, 2) H. J. Ryser, Combinatorial Mathematics, Carus Monographs, MAA, 1963.
[Gale57](1, 2) D. Gale, A theorem on flows in networks, Pacific J. Math. 7(1957)1073-1082.
sage.combinat.integer_vector.is_gale_ryser(r, s)

Tests whether the given sequences satisfy the condition of the Gale-Ryser theorem.

Given a binary matrix \(B\) of dimension \(n\times m\), the vector of row sums is defined as the vector whose \(i^{\mbox{th}}\) component is equal to the sum of the \(i^{\mbox{th}}\) row in \(A\). The vector of column sums is defined similarly.

If, given a binary matrix, these two vectors are easy to compute, the Gale-Ryser theorem lets us decide whether, given two non-negative vectors \(r,s\), there exists a binary matrix whose row/colum sums vectors are \(r\) and \(s\).

This functions answers accordingly.

INPUT:

  • r, s – lists of non-negative integers.

ALGORITHM:

Without loss of generality, we can assume that:

  • The two given sequences do not contain any \(0\) ( which would correspond to an empty column/row )
  • The two given sequences are ordered in decreasing order (reordering the sequence of row (resp. column) sums amounts to reordering the rows (resp. columns) themselves in the matrix, which does not alter the columns (resp. rows) sums.

We can then assume that \(r\) and \(s\) are partitions (see the corresponding class Partition)

If \(r^*\) denote the conjugate of \(r\), the Gale-Ryser theorem asserts that a binary Matrix satisfying the constraints exists if and only if \(s\preceq r^*\), where \(\preceq\) denotes the domination order on partitions.

EXAMPLES:

sage: from sage.combinat.integer_vector import is_gale_ryser
sage: is_gale_ryser([4,2,2],[3,3,1,1])
True
sage: is_gale_ryser([4,2,1,1],[3,3,1,1])
True
sage: is_gale_ryser([3,2,1,1],[3,3,1,1])
False

REMARK: In the literature, what we are calling a Gale-Ryser sequence sometimes goes by the (rather generic-sounding) term ‘’realizable sequence’‘.

sage.combinat.integer_vector.list2func(l, default=None)

Given a list l, return a function that takes in a value i and return l[i-1]. If default is not None, then the function will return the default value for out of range i’s.

EXAMPLES:

sage: f = sage.combinat.integer_vector.list2func([1,2,3])
sage: f(1)
1
sage: f(2)
2
sage: f(3)
3
sage: f(4)
Traceback (most recent call last):
...
IndexError: list index out of range
sage: f = sage.combinat.integer_vector.list2func([1,2,3], 0)
sage: f(3)
3
sage: f(4)
0

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