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### 3D generalization of Gaussian q-binomial coefficient

It is known that the coefficient of $q^t$ in Gaussian binomial coefficient $\binom{m+n}m_q$ equals the number of permutations of the multiset $\{0^m, 1^n\}$ with $t$ inversions. Is there a closed ...
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### Can I apply $q$-Lagrange Inversion formula?

Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this? Moreover, I have ...
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### Bijection between forests and skew SYT + Cyclic sieving

Consider the two-row skew shape $\lambda_n = (2n+1,n)/(1)$. The number of standard Young tableaux of this shape is $\binom{3n}{n}-\binom{3n}{n-2}$ (since one can easily biject this to the set of non-...
1 vote
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### Has anyone met this "$q$-character" table for $S_4$?

Is anyone aware of the following $q$-character table for the symmetric group $S_4$? \begin{array}{|c|c|c|c|c|c|} \hline \mathrm{conj}\backslash\mathrm{rep} & 2+1+1 & 3+1 & ...
91 views

### Addition formulas for q-analogs of trigonometric functions

Sine and Cosine functions possess notable formulas for addition of angles $$\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b).$$ One can ...
1 vote
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### Evaluate $\det[[\lfloor\frac{aj-(a+1)k}n\rfloor]_q]_{1\le j,k\le n}$ and $\det[[\lceil\frac{(a+1)j-ak}n\rceil]_q]_{1\le j,k\le n}$

The $q$-analogue of an integer $m$ is defined by $[m]_q=(1-q^m)/(1-q)$. Note that $\lim_{q\to1}[m]_q=m$. I have formulated the following conjecture on determinants involving the floor function and the ...
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### In search of a $q$-analogue of a Catalan identity

Let $C_n=\frac1{n+1}\binom{2n}n$ be the all-familiar Catalan numbers. Then, the following identity has received enough attention in the literature (for example, Lagrange Inversion: When and How): \...
128 views

### A recursion involving binomial coefficients: looking for a q-analog

Let $a_n := \frac{1}{2n+1}\binom{3n}{n}$. Then it is known that (one can find references in the OEIS for this.) $$a_n = \sum_{\substack{i,j,k \geq 0 \\ i+j+k=n-1} } a_i a_j a_k.$$ Is there a natural ...
164 views

Let $[n]_q=1+q+\dots +q^{n-1}$, ${[n]_q}! =_q _q \dots [n]_q$ and $\binom{n}{j}_q = \frac{[n]_q!}{[j]_q![n-j]_q!}$ be the usual $q$-notation. Consider the polynomials $p_n(q,r,x)= \sum_{j=0}^n ... 6 votes 0 answers 212 views ### Gaussian coefficients identity I am having difficulty showing the equivalence between (11) and (15) of Delsarte - Association schemes and$t$-designs in regular semilattices. It is somehow an application of Möbius inversion, but I ... 9 votes 2 answers 425 views ### Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra? In a Lie algebra$\mathfrak{g}$the Jacobi identity$\newcommand{\bracket}{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$holds. ... 3 votes 1 answer 161 views ### Is there a$q$-analogue to Shapiro's convolution identity? Let$C_n=\frac1{n+1}\binom{2n}n$denote the Catalan numbers. This question is motivated by the (unanswered) MO post by Alexander Burstein and my own (answered by Fedor Petrov) MO post. Specifically, ... 7 votes 1 answer 285 views ### Looking for a$q$-analogue of a binomial identity The following identity is well-known and there are a few proofs to it (see Bijective proof problems, by R P Stanley, for this and similar formulae): $$\sum_{k=0}^n\binom{2k}k\binom{2n-2k}{n-k}=4^n \... 6 votes 1 answer 236 views ### Enumerating subspaces of \mathbb{F}_q^n in terms of words and inversions When q is a prime power, then on the one hand the q-binomial coefficient \binom{n}{k}_q equals the number of k-dimensional subspaces of \mathbb{F}_q^n, and on the other hand it is the ... 5 votes 1 answer 160 views ### A q-analogue of a characterization of polynomials by binomial coefficients Considering the binomial coefficient \binom{x}{m} as a polynomial in x, the span of \binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d} is exactly the polynomials of degree \le d. A closely ... 19 votes 1 answer 604 views ### What is the groupoid cardinality of the category of vector spaces over a finite field? For any groupoid, it's groupoid cardinality is the sum of the reciprocals of the automorphism groups over the isomorphism classes. Let us consider the category of vector spaces over a finite field \... 17 votes 1 answer 820 views ### Proof of certain q-identity for q-Catalan numbers Let us use the standard notation for q-integers, q-binomials, and the q-analog$$ \operatorname{Cat}_q(n) := \frac{1}{[n+1]_q} \left[\matrix{2n \\ n}\right]_q. $$I want to prove that for all ... 6 votes 0 answers 235 views ### A recursion which defines polynomials with integer coefficients? Let [n]=1+q+\dots+q^{n-1} and u(n)=\prod_{j=1}^n \gcd([j],[n]). Define$$r(n)=\sum_{d|n,d>1}{(-1)^d \frac{u(n)}{du(\frac{n}{d})^d}r\Big(\frac{n}{d}\Big)^d}+\frac{(1-q)^{n-1}u(n)}{n[n]}$$with ... 8 votes 2 answers 476 views ### Relationship between q-Weyl dimension formula and q-analog of weight multiplicity? \DeclareMathOperator\dim{dim}For a dominant (integral) weight \lambda and any (integral) weight \mu of a simple Lie algebra \mathfrak{g}, Lusztig's q-analog of weight multiplicty K_{\lambda,... 9 votes 7 answers 682 views ### Important combinatorial and algebraic interpretations of the coefficients in the polynomial [n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1}) What are some important combinatorial and algebraic interpretations of the coefficients in the polynomial$$[n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1})?$$As motivation, I will give ... 11 votes 2 answers 532 views ### q-analogs of total positivity A real matrix M is called totally positive if all of its minors are positive; these matrices have been extensively studied, and there are generalizations to other Lie types, for example by Lusztig. ... 22 votes 2 answers 721 views ### A q-rious identity Let [x]_q=\frac{1-q^x}{1-q}, [n]_q!=_q_q\cdots[n]_q and {\binom{x}{n}}_{q}=\frac{[x]_q[x-1]_q\cdots[x-n+1]_q }{[n]_q!}. Computer experiments suggest that$$\det \left(q^\binom{i-j}{2}\... 6 votes 1 answer 342 views ### Lusztig's$q$-analog of weight multiplicity with product formula For partitions$\lambda, \mu \vdash n$, the Kostka-Foulkes polynomial$K_{\lambda,\mu}(q)$, a$q$-analog of the Kostka coefficient$K_{\lambda,\mu}$, has a combinatorial description, due to Lascoux ... 8 votes 1 answer 223 views ### Prominent examples of$q$-analogs without known cyclic sieving The cyclic sieving phenomenon is nicely summarized in the following AMS Notices "What is...?" article: https://www.ams.org/notices/201402/rnoti-p169.pdf. In that article, Reiner, Stanton, and White ... 8 votes 1 answer 274 views ### Product of$q$-analogues Background Recall that the$q$-analogue$[n]_q\in\mathbb Z[q]$of a natural number$n\in\mathbb N$is defined as $$[n]_q := \frac{q^n -1}{q-1}$$ the idea being that formulas involving$q$will ... 6 votes 0 answers 122 views ### Q-analogue of an inequality Pick integers$b\geq a \geq 0$and$k\geq j\geq 0$. It is not super-difficult to prove the inequality $$\binom{kb}{ka}^j \geq \binom{jb}{ja}^k.$$ This is actually quite a nice inequality that was ... 4 votes 0 answers 112 views ### Positivity of q-analogs of central binomial coefficients? With the usual$q-$notations$[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},[n]_q!=_q_q\cdots[n]_q$and$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$let $$b(n,k,r,q)=\det\left(q^{r\... 2 votes 2 answers 216 views ### q-factorial coefficient asymptotics Consider the [n]!_q = \prod\limits_{k = 1}^{n} \frac{q^k - 1}{q - 1} = \sum\limits_{k = 0}^{\binom n 2} c_k q^k and let \{f_n\}_{n \in \mathbb{N}} be the sequence of the functions on [0; 1] ... 6 votes 0 answers 189 views ### Conjecture for a certain Cauchy-type determinant Given the Cauchy-like matrix$$ \mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{ \Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right) }{ \Gamma(m)\,\Gamma(n) } \frac{m-\frac{3}{4}} {\... 2 votes 1 answer 224 views ### Major index generating polynomial for border-strip tableaux The Question in its original form has been answered, but there is a follow-up, see the end of the post. A border-strip is a skew Young diagram that does not contain a$2 \times 2$-box. A border-strip ... 6 votes 0 answers 188 views ### A curious$q$-identity Let$[x]_{q}=\frac{1-q^x}{1-q}$and$\binom{x}{n}_{q}$denote a$q$-binomial coefficient. Let$A_n(x,q)$be the$n\times n $matrix with entries $$q^\binom{i-j}{2}\binom{i+j+x}{i-j+1}_{q},$$$0 \le i,...
Let $q$ be a formal variable and for every positive integer $n$ let $$[n]_q! = 1 (1 + q)(1 + q + q^2) \dotsm (1 + q + \dotsb + q^{n-1})$$ be the $q$-factorial. Is it true that $[n]_q! + 1$ is an ...