Equal knapsack identities between symmetric group character degrees

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 $f^\lambda$ where $\lambda$ is a partition of $n$ into three parts of the same parity. In particular, the sum of the ``fat-hook'' degrees $f^{(k,k,1^{n-2k})}+f^{(k+1,k+1,1^{n-2k-2})}$ equals the sum of all $f^\lambda$ where $\lambda$ 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