David J. Hemmer, Armin Straub, Karlee J. Westrem — Preprint — 2025
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.Download
| Link | Size | Description | Hits |
|---|---|---|---|
| knapsack-syt.pdf | 408.30 KB | Preprint (PDF, 16 pages) | 27 |
BibTeX
@article{knapsack-syt-2025,
author = {David J. Hemmer and Armin Straub and Karlee J. Westrem},
title = {Equal knapsack identities between symmetric group character degrees},
journal = {Preprint},
year = {2025},
}