A $q$-analogue of Wilson's congruence

Abstract

Let ${\mathcal{C}}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\dots ,n\}$. Let $${\mathfrak{f}}_n(q):=\sum _{\sigma \in {\mathcal{C}}_{n+1}}{q}^{\mathrm{m}aj\sigma }$$ be a $q$-analogue of the factorial $n!$, where $\mathrm{m}aj$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $${\mathfrak{f}}_{n-1}(q)\equiv \mu (n)(\mathrm{mod}{\Phi }_n(q)),$$ where $\mu$ denotes the Möbius function and ${\Phi }_n(q)$ is the $n$-th cyclotomic polynomial.

This is a preliminary draft