Quotient of a partition ($Q_e(\lambda )$)

Fix an integer $e\ge 1$ and represent a partition of an integer $\lambda$ by a beta set $B$ with $m$ beads, where $m$ is a multiple of $e$ and at least the number of rows. On residue runner $r$, formEach $B_r$ is a beta set of a partition, decoding to ${\lambda }^{(r)}$. The ordered $e$-quotient is $Q_e(\lambda )=({\lambda }^{(0)},\dots ,{\lambda }^{(e-1)})$. Increasing $m$ by a multiple of $e$ leaves these partitions unchanged. Allowing arbitrary changes of the bead count can cyclically relabel the runners, so the bead-count convention matters for an ordered quotient.

For $\lambda =(3,1)$, use $e=m=4$, giving $B=\{6,3,1,0\}$. The runner sets are $\{0\},\{0\},\{1\},\{0\}$, hence $Q_4(3,1)=(\varnothing ,\varnothing ,(1),\varnothing )$. The weight of a partition is the sum of the quotient sizes: four cells are removed in reaching the empty core of a partition.

Hooks of $\lambda$ whose lengths are divisible by $e$ correspond bijectively to hooks in the $e$-quotient. A bead-gap pair on one runner with distance $eh$ becomes a bead-gap pair of distance $h$ in that runner partition, and hook removal commutes with this correspondence.