Core of a partition ($C_e(\lambda )$)

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