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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(...

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^{...

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 $...

Let $\mathfrak{S}_n$ be the permutation group on $\{1,\dots,n\}$. Given $\pi=\pi_1\pi_2\dots\pi_n\in\mathfrak{S}_n$, its major index statistic is denoted maj$(\pi)$. Define the polynomials $$Q_{n,k}(x)...