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The spt-crank for overpartitions

Frank G. GarvanChris Jennings-Shaffer — 2014

Acta Arithmetica

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions s p t ¯ ( n ) , s p t ¯ 1 ( n ) , s p t ¯ 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...

Self-conjugate vector partitions and the parity of the spt-function

George E. AndrewsFrank G. GarvanJie Liang — 2013

Acta Arithmetica

Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. Recently, we found new combinatorial interpretations of congruences for the spt-function modulo 5 and 7. These interpretations were in terms of a restricted set of weighted vector partitions which we call S-partitions. We prove that the number of self-conjugate S-partitions, counted with a certain weight, is related to the coefficients of a certain mock theta function studied by the first author,...

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