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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 ...

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 $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...

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,\...

We say two permutations $\pi_1$ and $\pi_2$ in the symmetric group $\mathfrak{S}_n$ are $k$-equivalent, denoted $\pi_1 \sim_k \pi_2$, if one can be determined from the other after a finite number of ...