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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 …

While looking at a representation theory question, I came up with the following sort of object. I want to know if it comes up often in combinatorics or some other area of mathematics. Let $P$ be a fi …

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 …

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), …

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^{ …

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 …

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 $g-I$ …

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 abelian group in question is the product of its Sylow-$p$ subgroups, which are preserved by automorphisms. Therefore the orbits in it are the products of orbits in the Sylow $p$-subgroups. Therefo …

Combinatorics: Theory and Applications by V. Krishnamurthy. This is an undergraduate text. Comprehensive and clear. If you live in India, there is an additional benefit: it can be yours for Rs. 275 …

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 …

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 …

Let's say you want to compute the Hall polynomial $g^\lambda_{(r),\mu}(p)$. According to [Dutta and Prasad, Degenerations and orbits in finite abelian groups], the orbits under the automorphism group …

The RS correspondence is a correspondence which associates to each permutation a pair of standard Young tableaux of the same shape. The RSK correspondence associates to each integer matrix (with non- …