= Power-core truncation of a prime-core tower
For a <prime number> $p$, integer $k\geq1$, 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 $0,\ldots,k-1$ 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>.
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