# Tag Info

Accepted

### A "quantum" identity: in search of a proof -Part II

Both sides are equal to $\binom{x+y+1}{n}_q$. This enumerates lattice paths in an $n\times (x+y-n+1)$ rectangle, according to the area statistic. We will assume that these paths start at $(0,0)$ and ...
Accepted

Accepted

### Schur-Weyl duality and q-symmetric functions

As Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those ...
• 2,991
Accepted

### Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do ...

### q-Catalan numbers from Grassmannians

There are a few nice answers to related questions. Unfortunately none of them quite answers the question you asked. The $q$-Catalan number $\frac{1}{[n+1]_q}{ 2n \brack n}_q$ is the Hilbert series of ...
• 3,541
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### Total positivity of $q$-Pascal matrix?

Yes. Consider the set $V$ of points with integer coordinates as vertices of a weighted directed acyclic graph. Namely, for any $(i,j)\in V$, the edge from $(i,j)$ to $(i+1,j)$ has weight 1, the edge ...
• 91.9k

• 122k

### A divisibility of q-binomial coefficients combinatorially

Assume $q$ is a prime power. Then there is a straightforward combinatorial interpretation. Consider the group action of $\mathbb F_{q^{a+b}}^\times / \mathbb F_{q^\times}$ on the space of $a$-...
• 122k
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### In search of a $q$-analogue of a Catalan identity

This identity is known as Jonah's formula (special case with $n\rightarrow 2n$ and $r\rightarrow n$, see "Catalan Numbers with Applications" by Thomas Koshy, pg. 325-326 for a combinatorial ...
Accepted

• 91.9k
Accepted

### Mysterious symmetry - in search for a bijection

Oliver Pechenik and I have some partial progress to report. Maybe someone else can see how to supply the remaining missing ingredients. First, let's establish some notation. We say your ascent ...
• 1,769

### A divisibility of q-binomial coefficients combinatorially

A proof of nonnegativity appears in https://arxiv.org/pdf/0912.1578.pdf. The number $\frac{1}{a+b}{a+b\choose a}$ is called a rational Catalan number by Drew Armstrong. See for instance http://www....
• 46.1k
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### Discriminants of some $q$-analogs of $(1+x)^n$

This is true. We have \begin{align*} p_n (q^{-1}, 1-r, x) &= \sum_{j=0}^n q^{ (r-1) \binom{j}{2}} \binom{n}{j}_{q^{-1}} x^j \\ &= \sum_{j=0}^n q^{ (r-1) \binom{j}{2}} q^{-j (n-j)} \binom{n}...
• 122k
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### What is the value of this sum involving q-binomials?

Doron Zeilberger has written a Maple code for checking and proving ordinary binomial identities and their $q$-analogues. What you need in the present case is the package called qEKHAD. I just tested ...
• 39.3k
Accepted

### Generating function for certain partitions (with a restriction on the Durfee square)

Lemma. Fix $n$ and $m$. Consider pairs of partitions $(\lambda,\mu)$ such that $\lambda$ has $m$ parts and $|\lambda|+|\mu|=n$. Let $A$ be the number of pairs for which $\max(\lambda)>\max(\mu)$ (...
• 91.9k
OK, here goes. We start with changing the notation ($z\to 20z^2$, $-z-3\to r$, $20rz\to y$ means that what was denoted by $z$ will be denoted by $20z^2$ from now on, $r$ is $-z-3$ with new $z$, so it ...