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Let $\mathfrak{S}_n$ denote the permutation group, and $I_0(n)=\sum_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions see A000085 for more interpretations. There is also these numbers $...

I came across this post Coefficients of the Inflated Eulerian Polynomial by AULI-GRAHAM-SAVAGE. In particular, the polynomials related to descents interested me $$P_n(x)=\sum_{\pi\in\mathfrak{S}_n}x^{...