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Some Parity Results for 16-Cores

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Abstract

Kolitsch and Sellers showed recently that a8(n), the number of 8-core partitions of n, is even when n belongs to certain arithmetic progressions. We prove a similar result for 16-cores. In doing so, we prove the surprising result that the a16(n), given by

$$\sum\limits_{n \geqslant 0} {a_{16} \left( n \right)q^n = \frac{{(q^{16} )_\infty ^{16} }} {{(q)_\infty }}} , $$

satisfy

$$a_{16} (43046721n + 457371400) \equiv a_{16} (n){\text{ (mod 2)}}{\text{.}} $$

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Authors and Affiliations

  1. School of Mathematics, UNSW, Sydney, 2052, NSW, Australia

    Michael D. Hirschhorn

  2. Department of Science and Mathematics, Cedarville College, P.O. Box 601, Cedarville, Ohio, 45314

    James A. Sellers

Authors
  1. Michael D. Hirschhorn
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  2. James A. Sellers
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Hirschhorn, M.D., Sellers, J.A. Some Parity Results for 16-Cores. The Ramanujan Journal 3, 281–296 (1999). https://doi.org/10.1023/A:1009879303577

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