For partitions λ and μ of the same integer, replacing each square by d^2 squares makes the character value χ_{λ̲}(μ̲) divisible by d!.
Almost all entries in the character table of the symmetric group are multiples of any given prime
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We show that almost every entry in the character table of $S_N$ is divisible by any fixed prime as $N\to\infty$. This proves a conjecture of Miller.
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Congruences in character tables of symmetric groups
For partitions λ and μ of the same integer, replacing each square by d^2 squares makes the character value χ_{λ̲}(μ̲) divisible by d!.