 Abacus of a partition

An $e$-runner abacus arranges the nonnegative integers by their residues modulo $e$ and places beads at the positions in a beta set. Moving a bead up one place on its runner removes an $e$-hook.

An abacus represents the beta set of a partition by beads at integer positions. Positions with the same residue modulo the chosen runner count lie on one runner; moving a bead to an empty position above it represents the corresponding rim-hook removal.