# A $q$-analogue of Wilson's congruence

## TL;DR

This paper establishes a $q$-analogue of Wilson's congruence involving permutation cycles, the major index, and cyclotomic polynomials, extending classical number theory results into the realm of $q$-series.

## Contribution

It introduces a novel $q$-analogue of Wilson's congruence, connecting permutation statistics with cyclotomic polynomials and the Möbius function.

## Key findings

- Proves a $q$-analogue of Wilson's congruence involving the major index.
- Establishes congruence relations modulo cyclotomic polynomials.
- Links permutation cycle enumeration with classical number theory concepts.

## Abstract

Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{\sigma\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,\sigma} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $$ {\mathfrak f}_{n-1}(q)\equiv\mu(n)\pmod{\Phi_n(q)}, $$ where $\mu$ denotes the M\"obius function and $\Phi_n(q)$ is the $n$-th cyclotomic polynomial.