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For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively. R Stanley proved the following ...

For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let $m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | \...

This is a 2nd installment to my earlier MO question for which Mark Wildon furnished a clean answer. $\mathcal{O}(\pi)$ and $\mathcal{E}(\pi)$ stand for the number of odd and even cycles of a ...

What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$? I ...