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!.
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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!.