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

George E. Andrews, Frank G. Garvan, and Jie Liang. "Self-conjugate vector partitions and the parity of the spt-function." Acta Arithmetica 158.3 (2013): 199-218. <http://eudml.org/doc/279005>.

@article{GeorgeE2013,
abstract = {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.},
author = {George E. Andrews, Frank G. Garvan, Jie Liang},
journal = {Acta Arithmetica},
keywords = {smallest parts function; spt-function; partitions; rank; Crank; vector partitions; Ramanujan's lost notebook; congruences; basic hypergeometric series; mock theta functions},
language = {eng},
number = {3},
pages = {199-218},
title = {Self-conjugate vector partitions and the parity of the spt-function},
url = {http://eudml.org/doc/279005},
volume = {158},
year = {2013},
}

TY - JOUR
AU - George E. Andrews
AU - Frank G. Garvan
AU - Jie Liang
TI - Self-conjugate vector partitions and the parity of the spt-function
JO - Acta Arithmetica
PY - 2013
VL - 158
IS - 3
SP - 199
EP - 218
AB - 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.
LA - eng
KW - smallest parts function; spt-function; partitions; rank; Crank; vector partitions; Ramanujan's lost notebook; congruences; basic hypergeometric series; mock theta functions
UR - http://eudml.org/doc/279005
ER -