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The spt-crank for overpartitions

100%
Garvan F., Jennings-Shaffer C.
Acta Arithmetica
|
2014
|
tom 166
|
nr 2
141-188
EN
Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\overline{spt}(n)$, $\overline{spt}_1(n)$, $\overline{spt}_{2}(n)$, 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 $M_2$-rank of a partition without repeated odd parts (Lovejoy and Osburn, 2009).
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Self-conjugate vector partitions and the parity of the spt-function

81%
Andrews G., Garvan F., Liang J.
Acta Arithmetica
|
2013
|
tom 158
|
nr 3
199-218
EN
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, Dyson and Hickerson. As a result we obtain an elementary q-series proof of Ono and Folsom's results for the parity of spt(n). A number of related generating function identities are also obtained.
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