A two-runner <partition abacus> gives
$$
Q_2(3,1)=((2),\varnothing),
\qquad
Q_2(2)=(\varnothing,(1)),
$$
and the 2-quotients of $(1)$ and $\varnothing$ are empty. Thus the <two-quotient tower of the partition three-one> has nonempty levels
$$
\boxed{
TQ_2(3,1)_0=((3,1)),
\quad TQ_2(3,1)_1=((2),\varnothing),
\quad TQ_2(3,1)_2=(\varnothing,(1),\varnothing,\varnothing).}
$$

Solution

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

{title2=$TQ_2(3,1)$}

The nonempty levels of the 2-quotient tower of $(3,1)$ are
$$
TQ_2(3,1)_0=((3,1)),\qquad
TQ_2(3,1)_1=((2),\varnothing),
$$
and
$$
TQ_2(3,1)_2=(\varnothing,(1),\varnothing,\varnothing).
$$


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 Incoming links: Two-quotient tower of the partition three-one