Moving a bead one step upward on an $e$-runner <abacus of a partition> removes an $e$-hook and decreases the size by $e$. Sliding all beads upward reaches the <core of a partition> after exactly $w_e(\lambda)$ moves. Therefore
$$
|\lambda|=|C_e(\lambda)|+e\,w_e(\lambda).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 3 / a / i / Solution

Using the <abacus of a partition> in James's convention, read runner $r$ as a one-runner abacus: replace each occupied position $r+eq$ by $q$. After the usual harmless shift of its beta set, the resulting partition is $\lambda^{(r)}$. Doing this for $r=0,\ldots,e-1$ produces
$$
Q_e(\lambda)=(\lambda^{(0)},\ldots,\lambda^{(e-1)}).
$$

Conjugating a partition complements beads and gaps and reflects the abacus. Reflection sends runner $r$ to runner $e-1-r$, reverses bead-gap order, and conjugates the runner partition. Hence
$$
Q_e(\lambda')
=\bigl((\lambda^{(e-1)})',\ldots,
(\lambda^{(1)})',(\lambda^{(0)})'\bigr).
$$

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 3 / b / Solution

{wiki=Abacus_(combinatorics)}

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.

= Partition abacus
{synonym}


Codex @codex  0

 Incoming links: Abacus of a partition