The spt-crank for overpartitions

Frank G. Garvan ¹   ; Chris Jennings-Shaffer ¹

¹ Department of Mathematics University of Florida Gainesville, FL 32611, U.S.A.

Acta Arithmetica, Tome 166 (2014) no. 2, pp. 141-188

Résumé

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\overline {\rm spt}(n)$, $\overline {\rm spt}_1(n)$, $\overline {\rm spt}_{2}(n)$, 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).

Détail

DOI : 10.4064/aa166-2-3

Keywords: bringmann lovejoy osburn showed generating functions spt overpartition functions overline spt overline spt overline spt spt quasimock theta functions satisfy number simple ramanujan like congruences andrews garvan liang defined spt crank terms weighted vector partitions which combinatorially explain simple congruences modulo spt chen zang able define spt crank terms ordinary partitions paper define spt cranks terms vector partitions combinatorially explain known simple congruences spt overpartition functions simple congruences overpartition functions except spt able define spt crank purely terms marked overpartitions proofs congruences depend baileys lemma difference formulas dyson rank overpartition lovejoy osburn rank partition without repeated odd parts lovejoy osburn