Solution
= Solution
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
$$
\boxed{Q_4(3,1)=(\varnothing,\varnothing,(1),\varnothing)}.
$$