 Residue content of a partition

The $e$-residue of a cell $(i,j)$ is $j-i$ modulo $e$, and the $e$-content is the multiset of cell residues. Removing an $e$-hook removes exactly one cell of each residue. Among partitions of a fixed size, two partitions have the same $e$-content exactly when they have the same $e$-core.