Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 3 / a

Moving a bead one step upward on an $e$-runner abacus of a partition removes an $e$-hook and decreases the size by $e$. Sliding all beads upward reaches the core of a partition after exactly $w_e(\lambda )$ moves. Therefore

The sum of the sizes of the partitions at the next level of the quotient tower of a partition isFor $e>1$, repeated passage to a quotient therefore makes every branch empty after finitely many levels. For $e=1$, the core is empty and $Q_1(\lambda )=(\lambda )$, so a nonempty partition repeats forever. Thus the tower has finite depth exactly when $e>1$ or $\lambda$ is empty.