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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An extension of Carlitz's bipartition identity


Author: George E. Andrews
Journal: Proc. Amer. Math. Soc. 63 (1977), 180-184
MSC: Primary 05A17
MathSciNet review: 0437350
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Abstract: Carlitz's bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function:

$\displaystyle \operatorname{smax}(n_1, n_2, \ldots, n_r) = n_1 + n_2 + \cdots + n_r - (r - 1)\min (n_1, n_2, \ldots, n_r).$

Let $ \pi_0(n_1, n_2, \ldots, n_r)$ denote the number of partitions of $ ({n_1},{n_2}, \ldots ,{n_r})$ in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let $ {\pi _1}({n_1},{n_2}, \ldots ,{n_r})$ denote the number of partitions of $ ({n_1},{n_2}, \ldots ,{n_r})$ in which each part has one of the $ 2r - 1$ forms: $ (a + 1,a,a, \ldots ,a),(a,a + 1,a, \ldots ,a), \ldots ,(a,a,a, \ldots ,a + 1),... ...a + 2),(ra + 3,ra + 3, \ldots ,ra + 3), \ldots ,(ra + r,ra + r, \ldots ,ra + r)$. Theorem: $ {\pi _0}({n_1}, \ldots ,{n_r}) = {\pi _1}({n_1}, \ldots ,{n_r})$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0437350-6
PII: S 0002-9939(1977)0437350-6
Keywords: Partitions, multipartite numbers
Article copyright: © Copyright 1977 American Mathematical Society