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

 

On certain weighted partitions and finite semisimple rings


Authors: L. B. Richmond and M. V. Subbarao
Journal: Proc. Amer. Math. Soc. 64 (1977), 13-19
MSC: Primary 10J20
MathSciNet review: 0439789
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Abstract: Let k be a fixed integer $ \geqslant 1$ and define $ {\tau _k}(n) = {\Sigma _{{d^k}/n}}1$. Thus $ {\tau _{1}}(n)$ is the ordinary divisor function and $ {\tau _k}(n)$ is the number of kth powers dividing n. We derive the asymptotic behaviour as $ n \to \infty $ of $ {P_k}(n)$ defined by

$\displaystyle \sum\limits_{n = 0}^\infty {{P_k}(n){x^n} = \prod\limits_{n = 1}^\infty {{{(1 - {x^n})}^{ - {\tau _k}(n)}}} .} $

Thus $ {P_k}(n)$ is the number of partitions of n where we recognize $ {\tau _k}(m)$ different colours of the integer m when it occurs as a summand in a partition. The case $ k = 2$ is of special interest since the number $ f(n)$ of semisimple rings with n elements when $ n = q_1^{{l_1}}q_2^{{l_2}} \ldots $ is given by $ f(n) = {P_2}({l_1}){P_2}({l_2}) \ldots$.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0439789-1
PII: S 0002-9939(1977)0439789-1
Article copyright: © Copyright 1977 American Mathematical Society