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The $q$-binomial theorem states that $$ \prod_{k=0}^{n-1}(1+q^kt) = \sum_{k=0}^n q^{\binom k2}{n\brack k}_q t^k. $$ This identity is a $q$-analogue of the binomial theorem $$ (1+t)^n = \sum_{k=0}^n \b …

The LGV lemma does not really require nodes to lie in a plane. In fact the statement is very simple without any geometric assumptions on the nodes. The role of the spatial arrangement of nodes is usua …

I know that the identity $$ s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i} $$ holds. Here $\alpha=1^{ …

For non-negative integers $k$ and $l$ let $p(k,l)$ denote the number of vector partitions of $(k,l)$. In other words, $p(k,l)$ is the number of ways of writing $$ (k,l) = (k_1,l_1)+\dotsb + (k_r,l_r), …

The $n\times n$ lattice is the set $X_n :=\{(i,j)\mid 1\leq i,j\leq n\}$, partially ordered by $(i,j)\leq (k,l)$ if $i\leq k$ and $j\leq l$. A linear extension of any poset $P$ (of cardinality $N$) i …

Define a standard bitableau of size $n$ to be a pair $(P_1, P_2)$ of standard tableaux of total size $n$ such that each of the integers $1,\dotsc, n$ occurs exactly once in either tableau. Then $I_2( …

Before getting into my answer, let me explain what I understand by a bitableau of size $n$ (I hope it is the same as what you mean). This is a pair of tableaux $(P, P')$, where each of the integers $1 …

The number of equivalence relations on a set of $n$ elements is the Bell number $B_n$. If we wish to count the number of equivalence classes on a set of $n$ elements where one of the classes is mark …

For Gaussian binomial coefficients we have $$ \sum_{k = 0}^n \binom nk_q = \sum_{m = 0}^\infty a_m q^m, $$ where $$ a_m = \sum_{\lambda\vdash m} \#\{k\in \mathbf Z_{\geq 0}\mid \lambda_1\leq n-k, \l …

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of refere …

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n( …

Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma, $$ |V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}. $$ Since $g-I$ …

It's all written up rather nicely in Heather Dornom's Honours thesis from 2005. She gives a version of the matrix ball construction that works in these cases and also explains growth models for the RS …

This question got answered by Gjergji Zaimi and Richard Stanley in the comments. I simply reproduce their comments here as an answer: A very simple explanation for this identity comes from the theory …

It is not difficult to show that the number of conjugacy classes in the alternating group $A_n$ is given by classes in the alternating group = no. of even partitions + no. of self-transpose partit …