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]}

References

1. George E. Andrews (2008). "A survey of multipartitions: congruences and identities". In Alladi, Krishnaswami (ed.). Surveys in Number Theory. Developments in Mathematics. Vol. 17. Springer-Verlag. pp. 1–19. ISBN   978-0-387-78509-7. Zbl   1183.11063.
2. Fayers, Matthew (2006). "Weights of multipartitions and representations of Ariki–Koike algebras". Advances in Mathematics . 206 (1): 112–144. CiteSeerX   10.1.1.538.4302 . doi: 10.1016/j.aim.2005.07.017 . Zbl   1111.20009.
3. Atkin, A. O. L. (1968). "Ramanujan congruences for ${\displaystyle p_{-k}(n)}$". Canadian Journal of Mathematics. 20: 67–78. doi: 10.4153/CJM-1968-009-6 .