Use four beads, for which $(3,1)$ has <beta set of a partition>[beta set]
$$
\{6,3,1,0\}.
$$
On runners of residues $0,1,2,3$, division by four gives respectively the beta sets
$$
\{0\},\quad\{0\},\quad\{1\},\quad\{0\}.
$$
Only $\{1\}$ represents a nonempty partition, namely $(1)$. Therefore the <four-quotient of the partition three-one> is
$$
\boxed{Q_4(3,1)=(\varnothing,\varnothing,(1),\varnothing)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 3 / c / i / Solution

{title2=$\beta(\lambda)$}

For $m\geq\ell(\lambda)$, a beta set is
$$
\{\lambda_i+m-i:1\leq i\leq m\}.
$$
A hook of length $h$ corresponds to a bead $b$ and an empty position $b-h$; removing the hook moves the bead to that gap.

= Beta set
{synonym}


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