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An $e^k$-runner abacus separates bead positions by their residue modulo $e^k$. Write such a residue in base $e$ asTaking one $e$-quotient sorts beads by $a_0$ and divides their positions by $e$; applying the operation again sorts by $a_1$, and so on. After $k$ stages, the iterated construction has selected exactly the same $e^k$ residue classes as the single $e^k$-quotient. The two conventional orderings may list the base-$e$ digits in opposite order, producing only a permutation of components.

Equivalently, induction on $k$ applies the same argument to every component of $Q_e(\lambda )$ and identifies the resulting $e^{k+1}$ runner partitions. Hence the iterated quotient equals a power quotient up to permutation statement is

For a prime number $p$, integer $k\ge 1$, and $\gamma =C_{p^k}(\lambda )$, taking the $p^k$-core of a partition preserves the entries of the core tower of a partition at levels $0,\dots ,k-1$ and makes every level at least $k$ empty. Also the weight of a partition $w_{p^k}(\lambda )$ equals the total size at level $k$ of the quotient tower of a partition.

For the first claim, a removable rim hook of length $p^k$ is a bead move by $p^k$ on the abacus of a partition. It preserves the numbers of beads on the $p$ runners and hence the $p$-core. Under the abacus divisible-hook correspondence, it becomes removal of a $p^{k-1}$-hook in one quotient component; iteration preserves the cores at the next $k-1$ levels. After all $p^k$-hooks are removed, every component at quotient level $k$ is empty, so every higher core level is empty as well. The weight assertion follows from iterated quotient equals a power quotient up to permutation and the size formula for the quotient of a partition.