Self-conjugate vector partitions and the parity of the spt-function
Volume 158 / 2013
Acta Arithmetica 158 (2013), 199-218
MSC: 05A17, 05A19, 11F33, 11P81, 11P82, 11P83, 11P84, 33D15.
DOI: 10.4064/aa158-3-1
Abstract
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.
