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

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