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Partitions of large multipartites with congruence conditions. I


Authors: M. M. Robertson and D. Spencer
Journal: Trans. Amer. Math. Soc. 219 (1976), 299-322
MSC: Primary 10J20
DOI: https://doi.org/10.1090/S0002-9947-1976-0401688-2
MathSciNet review: 0401688
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Abstract: Let $ p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ be the number of partitions of $ ({n_1}, \ldots ,{n_j})$ where, for $ 1 \leqslant l \leqslant j$, the lth component of each part belongs to the set $ {A_l} = \bigcup\nolimits_{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} } $ and $ M,q(l)$ and the $ {a_{lh(l)}}$ are positive integers such that $ 0 < {a_{l1}} < \cdots < {a_{lq(l)}} \leqslant M$. Asymptotic expansions for $ p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ are derived, when the $ {n_l} \to \infty $ subject to the restriction that $ {n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }$ for all l, where $ \in $ is any fixed positive number. The case $ M = 1$ and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case $ j = 1$ for different values of M.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0401688-2
Keywords: Partition, asymptotic, multipartite, congruence
Article copyright: © Copyright 1976 American Mathematical Society

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