A $q$-analogue of Wilson's congruence

Hao Pan, Yu-Chen Sun

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…