q-Analogues

sage.combinat.q_analogues.gaussian_binomial(n, k, q=None, algorithm='auto')

This is an alias of q_binomial().

See q_binomial() for the full documentation.

EXAMPLES:

sage: gaussian_binomial(4,2)
q^4 + q^3 + 2*q^2 + q + 1
sage.combinat.q_analogues.gaussian_multinomial(seq, q=None, binomial_algorithm='auto')

Return the \(q\)-multinomial coefficient.

This is also known as the Gaussian multinomial coefficient, and is defined by

\[\binom{n}{k_1, k_2, \ldots, k_m}_q = \frac{[n]_q!} {[k_1]_q! [k_2]_q! \cdots [k_m]_q!}\]

where \(n = k_1 + k_2 + \cdots + k_m\).

If \(q\) is unspecified, then the variable is the generator \(q\) for a univariate polynomial ring over the integers.

INPUT:

  • seq – an iterable of the values \(k_1\) to \(k_m\) defined above
  • q – (default: None) the variable \(q\); if None, then use a default variable in \(\ZZ[q]\)
  • binomial_algorithm – (default: 'auto') the algorithm to use in q_binomial(); see possible values there

ALGORITHM:

We use the equivalent formula

\[\binom{k_1 + \cdots + k_m}{k_1, \ldots, k_m}_q = \prod_{i=1}^m \binom{\sum_{j=1}^i k_j}{k_i}_q.\]

EXAMPLES:

sage: from sage.combinat.q_analogues import q_multinomial
sage: q_multinomial([1,2,1])
q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 2*q + 1
sage: q_multinomial([1,2,1], q=1) == multinomial([1,2,1])
True
sage: q_multinomial((3,2)) == q_binomial(5,3)
True
sage: q_multinomial([])
1
sage.combinat.q_analogues.q_binomial(n, k, q=None, algorithm='auto')

Return the \(q\)-binomial coefficient.

This is also known as the Gaussian binomial coefficient, and is defined by

\[\binom{n}{k}_q = \frac{(1-q^n)(1-q^{n-1}) \cdots (1-q^{n-k+1})} {(1-q)(1-q^2)\cdots (1-q^k)}.\]

See Wikipedia article Gaussian_binomial_coefficient.

If \(q\) is unspecified, then the variable is the generator \(q\) for a univariate polynomial ring over the integers.

INPUT:

  • n, k – the values \(n\) and \(k\) defined above
  • q – (default: None) the variable \(q\); if None, then use a default variable in \(\ZZ[q]\)
  • algorithm – (default: 'auto') the algorithm to use and can be one of the following:
    • 'auto' – automatically choose the algorithm; see the algorithm section below
    • 'naive' – use the naive algorithm
    • 'cyclotomic' – use cyclotomic algorithm

ALGORITHM:

The naive algorithm uses the product formula. The cyclotomic algorithm uses a product of cyclotomic polynomials (cf. [CH2006]).

When the algorithm is set to 'auto', we choose according to the following rules:

  • If q is a polynomial:

    When n is small or k is small with respect to n, one uses the naive algorithm. When both n and k are big, one uses the cyclotomic algorithm.

  • If q is in the symbolic ring, one uses the cyclotomic algorithm.

  • Otherwise one uses the naive algorithm, unless q is a root of unity, then one uses the cyclotomic algorithm.

EXAMPLES:

By default, the variable is the generator of \(\ZZ[q]\):

sage: from sage.combinat.q_analogues import q_binomial
sage: g = q_binomial(5,1) ; g
q^4 + q^3 + q^2 + q + 1
sage: g.parent()
Univariate Polynomial Ring in q over Integer Ring

The \(q\)-binomial coefficient vanishes unless \(0 \leq k \leq n\):

sage: q_binomial(4,5)
0
sage: q_binomial(5,-1)
0

Other variables can be used, given as third parameter:

sage: p = ZZ['p'].gen()
sage: q_binomial(4,2,p)
p^4 + p^3 + 2*p^2 + p + 1

The third parameter can also be arbitrary values:

sage: q_binomial(5,1,2) == g.subs(q=2)
True
sage: q_binomial(5,1,1)
5
sage: q_binomial(4,2,-1)
2
sage: q_binomial(4,2,3.14)
152.030056160000
sage: R = GF(25, 't')
sage: t = R.gen(0)
sage: q_binomial(6, 3, t)
2*t + 3

We can also do this for more complicated objects such as matrices or symmetric functions:

sage: q_binomial(4,2,matrix([[2,1],[-1,3]]))
[ -6  84]
[-84  78]
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.schur()
sage: q_binomial(4,1, s[2]+s[1])
s[] + s[1] + s[1, 1] + s[1, 1, 1] + 2*s[2] + 4*s[2, 1] + 3*s[2, 1, 1]
 + 4*s[2, 2] + 3*s[2, 2, 1] + s[2, 2, 2] + 3*s[3] + 7*s[3, 1] + 3*s[3, 1, 1]
 + 6*s[3, 2] + 2*s[3, 2, 1] + s[3, 3] + 4*s[4] + 6*s[4, 1] + s[4, 1, 1]
 + 3*s[4, 2] + 3*s[5] + 2*s[5, 1] + s[6]

TESTS:

One checks that the first two arguments are integers:

sage: q_binomial(1/2,1)
Traceback (most recent call last):
...
ValueError: arguments (1/2, 1) must be integers

One checks that \(n\) is nonnegative:

sage: q_binomial(-4,1)
Traceback (most recent call last):
...
ValueError: n must be nonnegative

This also works for variables in the symbolic ring:

sage: z = var('z')
sage: factor(q_binomial(4,2,z))
(z^2 + z + 1)*(z^2 + 1)

This also works for complex roots of unity:

sage: q_binomial(6,1,I)
1 + I

Check that the algorithm does not matter:

sage: q_binomial(6,3, algorithm='naive') == q_binomial(6,3, algorithm='cyclotomic')
True

One more test:

sage: q_binomial(4, 2, Zmod(6)(2), algorithm='naive')
5

REFERENCES:

[CH2006]William Y.C. Chen and Qing-Hu Hou, Factors of the Gaussian coefficients, Discrete Mathematics 306 (2006), 1446-1449. doi:10.1016/j.disc.2006.03.031

AUTHORS:

  • Frederic Chapoton, David Joyner and William Stein
sage.combinat.q_analogues.q_catalan_number(n, q=None)

Returns the \(q\)-Catalan number of index \(n\).

If \(q\) is unspecified, then it defaults to using the generator \(q\) for a univariate polynomial ring over the integers.

There are several \(q\)-Catalan numbers. This procedure returns the one which can be written using the \(q\)-binomial coefficients.

EXAMPLES:

sage: from sage.combinat.q_analogues import q_catalan_number
sage: q_catalan_number(4)
q^12 + q^10 + q^9 + 2*q^8 + q^7 + 2*q^6 + q^5 + 2*q^4 + q^3 + q^2 + 1
sage: p = ZZ['p'].0
sage: q_catalan_number(4,p)
p^12 + p^10 + p^9 + 2*p^8 + p^7 + 2*p^6 + p^5 + 2*p^4 + p^3 + p^2 + 1

The \(q\)-Catalan number of index \(n\) is only defined for \(n\) a nonnegative integer (trac ticket #11411):

sage: q_catalan_number(-2)
Traceback (most recent call last):
...
ValueError: Argument (-2) must be a nonnegative integer.
sage.combinat.q_analogues.q_factorial(n, q=None)

Returns the \(q\)-analogue of the factorial \(n!\).

If \(q\) is unspecified, then it defaults to using the generator \(q\) for a univariate polynomial ring over the integers.

EXAMPLES:

sage: from sage.combinat.q_analogues import q_factorial
sage: q_factorial(3)
q^3 + 2*q^2 + 2*q + 1
sage: p = ZZ['p'].0
sage: q_factorial(3, p)
p^3 + 2*p^2 + 2*p + 1

The \(q\)-analogue of \(n!\) is only defined for \(n\) a non-negative integer (trac ticket #11411):

sage: q_factorial(-2)
Traceback (most recent call last):
...
ValueError: Argument (-2) must be a nonnegative integer.
sage.combinat.q_analogues.q_int(n, q=None)

Return the \(q\)-analogue of the integer \(n\).

The \(q\)-analogue of the integer \(n\) is given by

\[\begin{split}[n]_q = \begin{cases} 1 + q + \cdots + q^{n-1}, & \text{if } n \geq 0, \\ -q^{-n} [-n]_q, & \text{if } n \leq 0. \end{cases}\end{split}\]

Consequently, if \(q = 1\) then \([n]_1 = n\) and if \(q \neq 1\) then \([n]_q = (q^n-1)/(q-1)\).

If the argument \(q\) is not specified then it defaults to the generator \(q\) of the univariate polynomial ring over the integers.

EXAMPLES:

sage: from sage.combinat.q_analogues import q_int
sage: q_int(3)
q^2 + q + 1
sage: q_int(-3)
(-q^2 - q - 1)/q^3
sage: p = ZZ['p'].0
sage: q_int(3,p)
p^2 + p + 1
sage: q_int(3/2)
Traceback (most recent call last):
...
ValueError: 3/2 must be an integer
sage.combinat.q_analogues.q_jordan(t, q)

INPUT:

  • \(t\) – a partition of an integer
  • \(q\) – an integer or an indeterminate

OUTPUT:

If \(q\) is the power of a prime number, the output is the number of complete flags in \(F_q^N\) (where \(N\) is the size of \(t\)) stable under a linear nilpotent endomorphism \(f\) whose Jordan type is given by \(t\), i.e. such that for all \(i\):

\[\dim (\ker f^i) = t[0] + \cdots + t[i-1]\]

If \(q\) is an indeterminate, the output is a polynomial whose values at powers of prime numbers are the previous numbers.

The result is cached.

EXAMPLES:

sage: from sage.combinat.q_analogues import q_jordan
sage: [q_jordan(mu,2) for mu in Partitions(5)]
[9765, 1029, 213, 93, 29, 9, 1]
sage: [q_jordan(mu,2) for mu in Partitions(6)]
[615195, 40635, 5643, 2331, 1491, 515, 147, 87, 47, 11, 1]

sage: q=PolynomialRing(ZZ,'q').gen()
sage: q_jordan(Partition([3,2,1]),q)
16*q^4 + 24*q^3 + 14*q^2 + 5*q + 1

If the partition is trivial (i.e. has only one part), we get the \(q\)-factorial (in this case, the nilpotent endomorphism is necessarily \(0\)):

sage: from sage.combinat.q_analogues import q_factorial
sage: q_jordan(Partition([5]),3) == q_factorial(5,3)
True
sage: q_jordan(Partition([11]),5) == q_factorial(11,5)
True

TESTS:

sage: q_jordan(Partition([4,3,1]),1)
Traceback (most recent call last):
...
ValueError: q must not be equal to 1

AUTHOR:

  • Xavier Caruso (2012-06-29)
sage.combinat.q_analogues.q_multinomial(seq, q=None, binomial_algorithm='auto')

Return the \(q\)-multinomial coefficient.

This is also known as the Gaussian multinomial coefficient, and is defined by

\[\binom{n}{k_1, k_2, \ldots, k_m}_q = \frac{[n]_q!} {[k_1]_q! [k_2]_q! \cdots [k_m]_q!}\]

where \(n = k_1 + k_2 + \cdots + k_m\).

If \(q\) is unspecified, then the variable is the generator \(q\) for a univariate polynomial ring over the integers.

INPUT:

  • seq – an iterable of the values \(k_1\) to \(k_m\) defined above
  • q – (default: None) the variable \(q\); if None, then use a default variable in \(\ZZ[q]\)
  • binomial_algorithm – (default: 'auto') the algorithm to use in q_binomial(); see possible values there

ALGORITHM:

We use the equivalent formula

\[\binom{k_1 + \cdots + k_m}{k_1, \ldots, k_m}_q = \prod_{i=1}^m \binom{\sum_{j=1}^i k_j}{k_i}_q.\]

EXAMPLES:

sage: from sage.combinat.q_analogues import q_multinomial
sage: q_multinomial([1,2,1])
q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 2*q + 1
sage: q_multinomial([1,2,1], q=1) == multinomial([1,2,1])
True
sage: q_multinomial((3,2)) == q_binomial(5,3)
True
sage: q_multinomial([])
1
sage.combinat.q_analogues.q_subgroups_of_abelian_group(la, mu, q=None, algorithm='birkhoff')

Return the \(q\)-number of subgroups of type mu in a finite abelian group of type la.

INPUT:

  • la – type of the ambient group as a Partition
  • mu – type of the subgroup as a Partition
  • q – (default: None) an indeterminat or a prime number; if None, this defaults to \(q \in \ZZ[q]\)
  • algorithm – (default: 'birkhoff') the algorithm to use can be one of the following:
    • 'birkhoff – use the Birkhoff formula from [Bu87]
    • 'delsarte' – use the formula from [Delsarte48]

OUTPUT:

The number of subgroups of type mu in a group of type la as a polynomial in q.

ALGORITHM:

Let \(q\) be a prime number and \(\lambda = (\lambda_1, \ldots, \lambda_l)\) be a partition. A finite abelian \(q\)-group is of type \(\lambda\) if it is isomorphic to

\[\ZZ / q^{\lambda_1} \ZZ \times \cdots \times \ZZ / q^{\lambda_l} \ZZ.\]

The formula from [Bu87] works as follows: Let \(\lambda\) and \(\mu\) be partitions. Let \(\lambda^{\prime}\) and \(\mu^{\prime}\) denote the conjugate partitions to \(\lambda\) and \(\mu\), respectively. The number of subgroups of type \(\mu\) in a group of type \(\lambda\) is given by

\[\prod_{i=1}^{\mu_1} q^{\mu^{\prime}_{i+1} (\lambda^{\prime}_i - \mu^{\prime}_i)} \binom{\lambda^{\prime}_i - \mu^{\prime}_{i+1}} {\mu^{\prime}_i - \mu^{\prime}_{i+1}}_q\]

The formula from [Delsarte48] works as follows: Let \(\lambda\) and \(\mu\) be partitions. Let \((s_1, s_2, \ldots, s_l)\) and \((r_1, r_2, \ldots, r_k)\) denote the parts of the partitions conjugate to \(\lambda\) and \(\mu\) respectively. Let

\[\mathfrak{F}(\xi_1, \ldots, \xi_k) = \xi_1^{r_2} \xi_2^{r_3} \cdots \xi_{k-1}^{r_k} \prod_{i_1=r_2}^{r_1-1} (\xi_1-q^{i_1}) \prod_{i_2=r_3}^{r_2-1} (\xi_2-q^{i_2}) \cdots \prod_{i_k=0}^{r_k-1} (\xi_k-q^{-i_k}).\]

Then the number of subgroups of type \(\mu\) in a group of type \(\lambda\) is given by

\[\frac{\mathfrak{F}(q^{s_1}, q^{s_2}, \ldots, q^{s_k})}{\mathfrak{F} (q^{r_1}, q^{r_2}, \ldots, q^{r_k})}.\]

EXAMPLES:

sage: from sage.combinat.q_analogues import q_subgroups_of_abelian_group
sage: q_subgroups_of_abelian_group([1,1], [1])
q + 1
sage: q_subgroups_of_abelian_group([3,3,2,1], [2,1])
q^6 + 2*q^5 + 3*q^4 + 2*q^3 + q^2
sage: R.<t> = QQ[]
sage: q_subgroups_of_abelian_group([5,3,1], [3,1], t)
t^4 + 2*t^3 + t^2
sage: q_subgroups_of_abelian_group([5,3,1], [3,1], 3)
144
sage: q_subgroups_of_abelian_group([1,1,1], [1]) == q_subgroups_of_abelian_group([1,1,1], [1,1])
True
sage: q_subgroups_of_abelian_group([5], [3])
1
sage: q_subgroups_of_abelian_group([1], [2])
0
sage: q_subgroups_of_abelian_group([2], [1,1])
0

TESTS:

Check the same examples with algorithm='delsarte':

sage: q_subgroups_of_abelian_group([1,1], [1], algorithm='delsarte')
q + 1
sage: q_subgroups_of_abelian_group([3,3,2,1], [2,1], algorithm='delsarte')
q^6 + 2*q^5 + 3*q^4 + 2*q^3 + q^2
sage: q_subgroups_of_abelian_group([5,3,1], [3,1], t, algorithm='delsarte')
t^4 + 2*t^3 + t^2
sage: q_subgroups_of_abelian_group([5,3,1], [3,1], 3, algorithm='delsarte')
144
sage: q_subgroups_of_abelian_group([1,1,1], [1], algorithm='delsarte') == q_subgroups_of_abelian_group([1,1,1], [1,1])
True
sage: q_subgroups_of_abelian_group([5], [3], algorithm='delsarte')
1
sage: q_subgroups_of_abelian_group([1], [2], algorithm='delsarte')
0
sage: q_subgroups_of_abelian_group([2], [1,1], algorithm='delsarte')
0

REFERENCES:

[Bu87](1, 2) Butler, Lynne M. A unimodality result in the enumeration of subgroups of a finite abelian group. Proceedings of the American Mathematical Society 101, no. 4 (1987): 771-775. doi:10.1090/S0002-9939-1987-0911049-8
[Delsarte48](1, 2) S. Delsarte, Fonctions de Mobius Sur Les Groupes Abeliens Finis, Annals of Mathematics, second series, Vol. 45, No. 3, (Jul 1948), pp. 600-609. http://www.jstor.org/stable/1969047

AUTHORS:

  • Amritanshu Prasad (2013-06-07): Implemented the Delsarte algorithm
  • Tomer Bauer (2013-09-26): Implemented the Birkhoff algorithm
sage.combinat.q_analogues.qt_catalan_number(n)

Returns the \(q,t\)-Catalan number of index \(n\).

EXAMPLES:

sage: from sage.combinat.q_analogues import qt_catalan_number
sage: qt_catalan_number(1)
1
sage: qt_catalan_number(2)
q + t
sage: qt_catalan_number(3)
q^3 + q^2*t + q*t^2 + t^3 + q*t
sage: qt_catalan_number(4)
q^6 + q^5*t + q^4*t^2 + q^3*t^3 + q^2*t^4 + q*t^5 + t^6 + q^4*t + q^3*t^2 + q^2*t^3 + q*t^4 + q^3*t + q^2*t^2 + q*t^3

The \(q,t\)-Catalan number of index \(n\) is only defined for \(n\) a nonnegative integer (trac ticket #11411):

sage: qt_catalan_number(-2)
Traceback (most recent call last):
...
ValueError: Argument (-2) must be a nonnegative integer.

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