= Quotient of a partition
{title2=$Q_e(\lambda)$}
= Partition quotient
{synonym}
Fix an <integer> $e\ge1$ 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 $r$, form
$$
B_r=\{(b-r)/e:b\in B,\ b\equiv r\pmod e\},\qquad 0\le r<e.
$$
Each $B_r$ is a <beta set of a partition>, decoding to $\lambda^{(r)}$. The ordered $e$-quotient is $Q_e(\lambda)=(\lambda^{(0)},\ldots,\lambda^{(e-1)})$. 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>.
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