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Partitions and Sylvester waves

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Abstract

The restricted partition function \(p_N(n)\) counts the partitions of the integer n into at most N parts. In the nineteenth century, Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as N and n both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to \(p_N(n)\) in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm.

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Notes

  1. As usual, the notation \(f(z)=O(g(z))\), or equivalently \(f(z) \ll g(z)\), means that there exists a C so that \(|f(z)|\leqslant C\cdot g(z)\) for all z in a specified range. The number C is called the implied constant.

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Authors and Affiliations

  1. Department of Mathematics, The Cuny Graduate Center, 365 Fifth Avenue, New York, NY, 10016-4309, USA

    Cormac O’Sullivan

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  1. Cormac O’Sullivan
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Correspondence to Cormac O’Sullivan.

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This research was supported, in part, by a grant of computer time from the City University of New York High Performance Computing Center under NSF Grants CNS-0855217, CNS-0958379, and ACI-1126113. Support for this project was also provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

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O’Sullivan, C. Partitions and Sylvester waves. Ramanujan J 47, 339–381 (2018). https://doi.org/10.1007/s11139-017-9939-9

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  • DOI: https://doi.org/10.1007/s11139-017-9939-9

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