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

George E. Andrews ¹   ; Frank G. Garvan ²   ; Jie Liang ²

¹ Department of Mathematics The Pennsylvania State University University Park, PA 16802, U.S.A.
² Department of Mathematics University of Florida Gainesville, FL 32611-8105, U.S.A.

Acta Arithmetica, Tome 158 (2013) no. 3, pp. 199-218

Résumé

Let ${\rm spt}(n)$ 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 $ {\rm spt}(n)$. A number of related generating function identities are also obtained.

Détail

DOI : 10.4064/aa158-3-1

Keywords: spt denote total number appearances smallest parts partitions recently found combinatorial interpretations congruences spt function modulo these interpretations terms restricted set weighted vector partitions which call s partitions prove number self conjugate s partitions counted certain weight related coefficients certain mock theta function studied first author dyson hickerson result obtain elementary q series proof ono folsoms results parity spt number related generating function identities obtained