Abacus of a partition

An $e$-runner abacus arranges the nonnegative integers by their residues modulo $e$ and places beads at the positions in a beta set. Moving a bead up one place on its runner removes an $e$-hook.

The $e$-core is obtained by repeatedly removing hooks of length $e$. On an $e$-runner abacus it is obtained by sliding every bead as high as possible, which proves independence of the order of removals.

The $e$-weight is the number of $e$-hooks removed to reach the $e$-core. It satisfiesand equals the total size of the partitions in the $e$-quotient.

Removing a rim hook of length two preserves the number of odd hook lengths minus the number of even hook lengths. If the 2-core is the staircase $(r,r-1,\dots ,1)$, every one of its $r(r+1)/2$ hooks is odd. Hence the difference for the original partition is

The $e$-quotient $Q_e(\lambda )=({\lambda }^{(0)},\dots ,{\lambda }^{(e-1)})$ is the tuple of partitions represented by the individual runners after their positions are divided by $e$.

Hooks of $\lambda$ whose lengths are divisible by $e$ correspond bijectively to hooks in the $e$-quotient. A bead-gap pair on one runner with distance $eh$ becomes a bead-gap pair of distance $h$ in that runner partition, and hook removal commutes with this correspondence.

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.

The $p$-core tower places at level $r$ the $p$-cores of all partitions at level $r$ of the quotient tower of a partition. If $q_r$ is the sum of the sizes at quotient level $r$ and $c_r$ the sum at core level $r$, then

If $n=|\lambda |$ and $d_p(n)$ is its base-$p$ digit sum, thenThis combines the Hook-length formula, Legendre formula, the abacus divisible-hook correspondence, and the recurrence $q_r=c_r+p{q}_{r+1}$.

For every partition $\lambda$,The core-tower formula reduces this to subadditivity of the base-$p$ digit sum across the sizes of the quotient partitions.

The $e$-residue of a cell $(i,j)$ is $j-i$ modulo $e$, and the $e$-content is the multiset of cell residues. Removing an $e$-hook removes exactly one cell of each residue. Among partitions of a fixed size, two partitions have the same $e$-content exactly when they have the same $e$-core.