Core Partition

A core partition with modulus , or -core partition, is a partition none of whose hook lengths is divisible by the positive integer . The hook length of a box in its Ferrers diagram is one plus the number of boxes to its right in the same row and below it in the same column. A simultaneous -core partition is both an -core and a -core.

For example, has hook lengths 3, 1, and 1, so it is a 2-core but not a 3-core. For relatively prime positive integers , the number of simultaneous core partitions, including the empty partition, is

(Anderson 2002). For , , the two are the empty partition and .

A corner is a box whose deletion still leaves a Ferrers diagram. Its count is the number of distinct parts. Cho et al. (2026) give formulas and bijections for self-conjugate partitions that are simultaneously -core, refined by their number of corners.