On the Divisibility of Character Values of the Symmetric Group

Fix a partition $\mu =({\mu }_1,\dots ,{\mu }_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let ${\chi }_{\mu }^{\lambda }$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type $({\mu }_1,\dots ,{\mu }_m,1^{n-k})$. We show that the proportion of partitions $\lambda$ of $n$ such that ${\chi }_{\mu }^{\lambda }$ is divisible by $d$ approaches $1$ as $n$ approaches infinity.