Congruences for the cycle indicator of the symmetric group

TL;DR

This paper extends Carlitz's congruence for cycle indicators of symmetric groups to a broader modulus and applies this to prove a conjecture related to Meixner polynomials, advancing understanding in algebraic combinatorics.

Contribution

It generalizes Carlitz's congruence for cycle indicators to a larger modulus for primes not equal to 2, and uses this to prove Junod's conjecture for Meixner polynomials.

Findings

Extended Carlitz's congruence to modulus npZ_p for p ≠ 2.

Proved Junod's conjecture for Meixner polynomials.

Enhanced understanding of symmetric group cycle indicators.

Abstract

Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a non negative integer, then we have the congruence $C_{r+np}\equiv (X_1^p-X_p)^nC_r \mod{pZ_p[X_1,\cdots,X_{r+np}]},$ where $Z_p$ is the ring of $p$-adic integers. We prove that for $p\neq 2$, the preceding congruence holds modulo $npZ_p[X_1,\cdots,X_{r+np}]$. This allows us to prove a Junod's conjecture for Meixner polynomials.