Congruences for the cycle indicator of the symmetric group

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 $C_{r+np}\equiv (X_1^p-X_p{)}^n{C}_r\mathrm{m}\mathrm{o}\mathrm{d}p{Z}_p[X_1,\cdots ,X_{r+np}],$ where $Z_p$ is the ring of $p$-adic integers. We prove that for $p\mathrm{\ne }2$, the preceding congruence holds modulo $np{Z}_p[X_1,\cdots ,X_{r+np}]$. This allows us to prove a Junod’s conjecture for Meixner polynomials.