# Multipartition

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In [number theory](/source/number_theory) and [combinatorics](/source/combinatorics), a '''multipartition''' of a positive [integer](/source/integer) ''n'' is a way of writing ''n'' as a [sum](/source/summation), each element of which is in turn an [integer partition](/source/integer_partition).<ref name="Andrews2008">{{cite book | editor1-first=Krishnaswami | editor1-last=Alladi |editor1-link= Krishnaswami Alladi | title=Surveys in Number Theory | series=Developments in Mathematics | volume=17 | publisher=[Springer-Verlag](/source/Springer-Verlag) | year=2008 | isbn=978-0-387-78509-7 | author=George E. Andrews | authorlink=George Andrews (mathematician) | chapter=A survey of multipartitions: congruences and identities | pages=1–19 | zbl=1183.11063}}</ref> The concept is also found in the theory of [Lie algebra](/source/Lie_algebra)s.<ref name="Andrews2008"/><ref>{{cite journal | journal=[Advances in Mathematics](/source/Advances_in_Mathematics) | volume=206 | issue=1 | year=2006 | pages=112–144 | title=Weights of multipartitions and representations of Ariki–Koike algebras | first=Matthew | last=Fayers | doi=10.1016/j.aim.2005.07.017 | doi-access=free | zbl=1111.20009 | citeseerx=10.1.1.538.4302 }}</ref>

==r-component multipartitions==
An ''r''-component multipartition of an integer ''n'' is an ''r''-tuple of partitions ''λ''<sup>(1)</sup>, ..., ''λ''<sup>(r)</sup> where each ''λ''<sup>(''i'')</sup> is a partition of some ''a''<sub>''i''</sub> and the  ''a''<sub>''i''</sub> sum to ''n''.  The number of ''r''-component multipartitions of ''n'' is denoted ''P''<sub>''r''</sub>(''n'').  Congruences for the function ''P''<sub>''r''</sub>(''n'') have been studied by [A. O. L. Atkin](/source/A._O._L._Atkin).<ref name="Andrews2008"/><ref>{{cite journal
| last1=Atkin | first1=A. O. L. | authorlink1=A. O. L. Atkin
| title=Ramanujan congruences for <math>p_{-k}(n)</math>
| journal=Canadian Journal of Mathematics
| volume=20
| date=1968
| pages=67-78
| doi=10.4153/CJM-1968-009-6 | doi-access=free}}</ref>

==References==

{{reflist}}

Category:Number theory
Category:Combinatorics

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