A survey of multipartitions: Congruences and identities.

*(English)* Zbl 1183.11063
Alladi, Krishnaswami (ed.), Surveys in number theory. New York, NY: Springer (ISBN 978-0-387-78509-7/hbk). Developments in Mathematics 17, 1-19 (2008).

Summary: The concept of a multipartition of a number, which has proved so useful in the study of Lie algebras, is studied for its own intrinsic interest. Following up on the work of Atkin, we present an infinite family of congruences for ${P}_{k}\left(n\right)$, the number of $k$-component multipartitions of $n$. We also examine the enigmatic tripentagonal number theorem and show that it implies a theorem about tripartitions. Building on this latter observation, we examine a variety of multipartition identities connecting them with mock theta functions and the Rogers-Ramanujan identities.

##### MSC:

11P81 | Elementary theory of partitions |

11P83 | Partitions: congruences and congruential restrictions |

05A30 | $q$-calculus and related topics |