Multipartition

In number theory and combinatorics, a multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn an integer partition.^[1] The concept is also found in the theory of Lie algebras.^[1][2]

r-component multipartitions

An r-component multipartition of an integer n is an r-tuple of partitions λ⁽¹⁾, ..., λ^(r) where each λ⁽ⁱ⁾ is a partition of some a_i and the a_i sum to n. The number of r-component multipartitions of n is denoted P_r(n). Congruences for the function P_r(n) have been studied by A. O. L. Atkin.^[1][3]