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 a partition. The concept is also found in the theory of Lie algebras.

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.

References

• George E. Andrews (2008). "A survey of multipartitions". in Alladi, Krishnaswami. Surveys in Number Theory. Developments in Mathematics. 17. Springer-Verlag. pp. 1–19. ISBN 978-0-387-78509-7. 
• Fayers, Matthew (2006). "Weights of multipartitions and representations of Ariki–Koike algebras". Advances in Mathematics 206 (1): 112–144. doi:10.1016/j.aim.2005.07.017.