Equal knapsack identities between symmetric group character degrees
David J. Hemmer, Armin Straub, Karlee J. Westrem
TL;DR
The paper investigates sums of irreducible character degrees $f^\lambda$ of the symmetric group $\Sigma_n$ over carefully chosen sets of partitions, revealing knapsack-type equalities that refine known Riordan-number interpretations. Central to the work is a refinement of the equality $\sum_{\lambda \in X} f^\lambda = R(n)$ by partitioning the index set into $X_1(n,k)$ and $X_2(n,k)$ and showing that each prescribed sum equals a fixed two-term partition-degree expression $f^{(k,k,1^{n-2k})}+f^{(k+1,k+1,1^{n-2k-2})}$ (with a complementary variant when parity conditions differ). The proofs combine the branching rule for $\Sigma_n$ with two auxiliary fixed-length identities obtained from the hook-length formula, and are extended by a broader analytic framework for odd-depth sums $L_{2d+1}(k,m)$, yielding alternative proofs and generalizations. The results connect to Riordan/Motzkin interpretations and prompt questions about potential bijective proofs and representation-theoretic interpretations, including links to modular phenomena and Regev-type identities.
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 $f^λ$ where $λ$ is a partition of $n$ into three parts of the same parity. In particular, the sum of the ``fat-hook'' degrees $f^{(k,k,1^{n-2k})}+f^{(k+1,k+1,1^{n-2k-2})}$ equals the sum of all $f^λ$ 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