Equal knapsack identities between symmetric group character degrees

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,1ⁿ⁻²ᵏ)+f^(k+1,k+1,1ⁿ⁻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