On the Divisibility of Character Values of the Symmetric Group

Fix a partition mu = (mu(1), ..., mu(m)) of an integer k and positive integer d. For each n >= k, let chi(lambda)(mu) denote the value of the irreducible character chi(lambda) of S-n, corresponding to a partition lambda of n, at a permutation with cycle type (mu(1), ..., mu(m) 1(n-k)). We show that the proportion of partitions lambda of n such that chi(lambda)(mu) is divisible by d approaches 1 as n approaches infinity.