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The spt-crank for overpartitions

sumproduct 添加于 2018/3/27 15:24:46  1083次阅读 | 0次推荐 | 0个评论

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions spt−−−(n), spt−−−1(n), spt−−−2(n), and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions. In this paper we define spt-cranks in terms of vector partitions that combinatorially explain the known simple congruences for all the spt-overpartition functions as well as new simple congruences. For all the overpartition functions except M2spt(n) we are able to define the spt-crank purely in terms of marked overpartitions. The proofs of the congruences depend on Bailey's Lemma and the difference formulas for the Dyson rank of an overpartition (Lovejoy and Osburn, 2008) and the M2-rank of a partition without repeated odd parts (Lovejoy and Osburn, 2009).

作 者:Frank G. Garvan, Chris Jennings-Shaffer
期刊名称: Acta Arithmetica
期卷页: 2014 第166卷 第141期 188页
学科领域:数理科学 » 数学 » 组合数学
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原文链接:https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/166/2/83471/the-spt-crank-for-overpartitions
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