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I was doing some counting in $S_n$ symmetric group I encountered the following problem, which also someway related to central factorial number. So given a $n$ cycle say $(1,2,\ldots,n)$, what are the ...

Here is an olympiad-level problem on elementary number theory: Let $a$ be an integer and $n$ a positive integer. Prove that \begin{align} \left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(...

This is a crosspost of a question I asked on Mathematics SE four months ago. Periodically bumping it and placing a bounty on it to attract more attention were to no avail. There are some comments ...

Let $(n),\mu$ be the partition of $n$ define $H_g^{m}((n);\mu)$ count's the number of tuples $(\tau_1,\ldots,\tau_r)$ of transposition in symmetric group $S_n$ with the following conditions $$ (1,2,\...