 Solution

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.