Quotient tower of a partition ($T{Q}_e(\lambda )$)

The $e$-quotient tower recursively takes the $e$-quotient of every partition at the preceding level. For $e>1$, the sum of the sizes at each new level is at most $1/e$ times the preceding sum, so every partition has finite depth. For $e=1$, the quotient is the original partition and the tower has finite depth only for the empty partition.

With runners ordered by residues $0,1,2,3$, an abacus for $(3,1)$ gives

The nonempty levels of the 2-quotient tower of $(3,1)$ areand

For every $e,k\ge 1$, the $e^k$-quotient $Q_{e^k}(\lambda )$ is a permutation of level $k$ of the $e$-quotient tower. Writing a runner residue modulo $e^k$ in base $e$ shows that taking one quotient chooses one digit at a time; iteration may reverse the order of those digits but selects the same runner partitions.