# 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. Zbl 1183.11063. - 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. Zbl 1111.20009.

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