The spt-crank for overpartitions

• Autorzy: Frank G. Garvan , Chris Jennings-Shaffer
• Języki publikacji: EN

Abstrakty

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).

Kategorie tematyczne

• 11P83: Partitions; congruences and congruential restrictions
• 05A19: Combinatorial identities, bijective combinatorics
• 11P82: Analytic theory of partitions
• 05A17: Partitions of integers
• 11F03: Modular and automorphic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 166
• Numer: 2
• Strony: 141-188

Daty

wydano

2014

Twórcy

autor

Frank G. Garvan

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

autor

Chris Jennings-Shaffer

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

Identyfikatory

DOI

10.4064/aa166-2-3