Equal knapsack identities between symmetric group character degrees

Document Type

Article

Publication Date

12-15-2026

Abstract

We prove a series of “knapsack” type equalities for irreducible character degrees of symmetric groups. That is, we find disjoint subsets of the partitions of n so that the two corresponding character-degree sums are equal. Our main result refines our recent description of the Riordan numbers as the sum of all character degrees fλ where λ is a partition of n into three parts of the same parity. In particular, the sum of the “fat-hook” degrees f(k,k,1n−2k)+f(k+1,k+1,1n−2k−2) equals the sum of all fλ where λ has three parts, with the second equal to k and the second and third of equal parity. We further prove an infinite family of additional “knapsack” identities between character degrees.

Publication Title

Journal of Algebra

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