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Consider the following refinement of permutation statistics. For $π ∈ S_n$, let: $\mathrm{fix}(π) = |\{i : π(i) = i\}|$ (number of fixed points) $\mathrm{exc}(π) = |\{i : π(i) > i\}|$ (number of ...

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 ...

How do you show that number of permutations of $\{1,2,3,\ldots,n\}$ such that image of no two consecutive numbers is consecutive is $$n! + \sum_{k = 1}^{n}(-1)^k\sum_{i = 1}^{k}\dbinom{k - 1}{i - 1}\...

I want to count the number of permutations of the given order $k$ in $S_n\;(\sigma^k=id,\sigma^l\neq id\;for\;l<k)$. I found some works about that problem, but they are more general than necessary. ...

Given a Ferrers board of shape $(b_1,\ldots,b_m)$, we define $r_k$ as number of ways to place $k$ non-attacking rooks (as in Chess). In section 2.4 of Stanley's Enumerative Combinatorics (vol. 1) it's ...