The expected number of parts in a partition ofn

Forn a positive integer letp(n) denote the number of partitions ofn into positive integers and letp(n,k) denote the number of partitions ofn into exactlyk parts. Let\(P(n) = \sum\limits_{k = 1}^\infty {kp(n,k)} \), thenP(n) represents the total number of parts in all the partitions ofn. In this paper we obtain the following asymptotic formula for\({{P(n)} \mathord{\left/ {\vphantom {{P(n)} {p(n)}}} \right. \kern-\nulldelimiterspace} {p(n)}}:{{P(n)} \mathord{\left/ {\vphantom {{P(n)} {p(n)}}} \right. \kern-\nulldelimiterspace} {p(n)}} = \sqrt {{{3n} \mathord{\left/ {\vphantom {{3n} {2\pi }}} \right. \kern-\nulldelimiterspace} {2\pi }}} (\log n + 2\gamma - \log {\pi \mathord{\left/ {\vphantom {\pi 6}} \right. \kern-\nulldelimiterspace} 6}) + 0(\log ^3 n).\).

Authors and Affiliations

1. Southern Illinois University, 62026, Edwardsville, IL, USA

   I. Kessler &  M. Livingston

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