Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 3 / b / Solution

LetA direct comparison of the affected bead-gap pairs shows that removing a rim 2-hook preserves $D$: the pairs whose parities change cancel in odd-even pairs. Repeating this removal givesEvery 2-core is a staircaseAll hook lengths in this staircase are odd, and it has $1+2+\cdots +r=r(r+1)/2$ cells. Hence the odd-minus-even hook count of a partition isThus the requested integer is $m=r+1=\ell (C_2(\lambda ))+1$.