Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

The spt-crank of a vector partition, or an S  -partition, was introduced by Andrews, Garvan and Liang. Let NS(m,n)NS(m,n) denote the net number of S-partitions of n with spt-crank m, that is, the number of S  -partitions (π1,π2,π3)(π1,π2,π3) of n with spt-crank m   such that the length of π1π1 is odd minus the number of S  -partitions (π1,π2,π3)(π1,π2,π3) of n with spt-crank m   such that the length of π1π1 is even. Andrews, Dyson and Rhoades conjectured that {NS(m,n)}m{NS(m,n)}m is unimodal for any n  , and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions, which implies NS(m,n)≥NS(m+1,n)NS(m,n)≥NS(m+1,n) for sufficiently large n and fixed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.