On the Divisibility of Character Values of the Symmetric Group

Fix a partition $μ=(μ_1,\dotsc,μ_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let $χ^λ_μ$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type $(μ_1,\dotsc,μ_m,1^{n-k})$. We show that the proportion of partitions $λ$ of $n$ such that $χ^λ_μ$ is divisible by $d$ approaches $1$ as $n$ approaches infinity.

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