Core tower of a partition ($T{C}_p(\lambda )$)

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.