# $$q$$-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

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

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]


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:

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

Return 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=None)

Return the $$q$$-Jordan number of $$t$$.

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

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

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

The result is cached.

INPUT:

• t – an integer partition, or an argument accepted by Partition
• q – (default: None) the variable $$q$$; if None, then use a default variable in $$\ZZ[q]$$

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_jordan([3,2,1])
16*q^4 + 24*q^3 + 14*q^2 + 5*q + 1
sage: q_jordan([2,1], x)
2*x + 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([5]) == q_factorial(5)
True
sage: q_jordan([11], 5) == q_factorial(11, 5)
True


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_pochhammer(n, a, q=None)

Return the $$q$$-Pochhammer $$(a; q)_n$$.

The $$q$$-Pochhammer symbol is defined by

$(a; q)_n = \prod_{k=0}^{n-1} (1 - aq^k)$

with $$(a; q)_0 = 1$$ for all $$a, q$$ and $$n \in \NN$$. By using the identity

$(a; q)_n = \frac{(a; q)_{\infty}}{(aq^n; q)_{\infty}},$

we can extend the definition to $$n < 0$$ by

$(a; q)_n = \frac{1}{(aq^n; q)_{-n}} = \prod_{k=1}^{-n} \frac{1}{1 - a/q^k}.$

EXAMPLES:

sage: from sage.combinat.q_analogues import q_pochhammer
sage: q_pochhammer(3, 1/7)
6/343*q^3 - 6/49*q^2 - 6/49*q + 6/7
sage: q_pochhammer(3, 3)
-18*q^3 + 6*q^2 + 6*q - 2
sage: q_pochhammer(3, 1)
0

sage: R.<q> = ZZ[]
sage: q_pochhammer(4, q)
q^10 - q^9 - q^8 + 2*q^5 - q^2 - q + 1
sage: q_pochhammer(4, q^2)
q^14 - q^12 - q^11 - q^10 + q^8 + 2*q^7 + q^6 - q^4 - q^3 - q^2 + 1
sage: q_pochhammer(-3, q)
1/(-q^9 + q^7 + q^6 + q^5 - q^4 - q^3 - q^2 + 1)


REFERENCES:

sage.combinat.q_analogues.q_stirling_number1(n, k, q=None)

Return the (unsigned) $$q$$-Stirling number of the first kind.

This is a $$q$$-analogue of sage.combinat.combinat.stirling_number1() .

INPUT:

• n, k – integers with 1 <= k <= n
• q – optional variable (default $$q$$)

OUTPUT: a polynomial in the variable $$q$$

These polynomials satisfy the recurrence

$s_{n,k} = s_{n-1,k-1} + [n-1]_q s_{n-1, k}.$

EXAMPLES:

sage: from sage.combinat.q_analogues import q_stirling_number1
sage: q_stirling_number1(4,2)
q^3 + 3*q^2 + 4*q + 3

sage: all(stirling_number1(6,k) == q_stirling_number1(6,k)(1)
....:     for k in range(1,7))
True

sage: x = polygen(QQ['q'],'x')
sage: S = sum(q_stirling_number1(5,k)*x**k for k in range(1, 6))
sage: factor(S)
x * (x + 1) * (x + q + 1) * (x + q^2 + q + 1) * (x + q^3 + q^2 + q + 1)


REFERENCES:

sage.combinat.q_analogues.q_stirling_number2(n, k, q=None)

Return the (unsigned) $$q$$-Stirling number of the second kind.

This is a $$q$$-analogue of sage.combinat.combinat.stirling_number2().

INPUT:

• n, k – integers with 1 <= k <= n
• q – optional variable (default $$q$$)

OUTPUT: a polynomial in the variable $$q$$

These polynomials satisfy the recurrence

$S_{n,k} = q^{k-1} S_{n-1,k-1} + [k]_q s_{n-1, k}.$

EXAMPLES:

sage: from sage.combinat.q_analogues import q_stirling_number2
sage: q_stirling_number2(4,2)
q^3 + 3*q^2 + 3*q

sage: all(stirling_number2(6,k) == q_stirling_number2(6,k)(1)
....:     for k in range(7))
True


REFERENCES:

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 indeterminate 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


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 Möbius 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, 2018): Implemented the Birkhoff algorithm and refactoring
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.