  The notion of the spt-crank of a vector partition, or an $S$-partition, was
introduced by Andrews, Garvan and Liang. Let $N_S(m,n)$ denote the number of
$S$-partitions of $n$ with spt-crank $m$. Andrews, Dyson and Rhoades
conjectured that $\{N_S(m,n)\}_m$ is unimodal for any $n$, and they showed that
this conjecture is equivalent to an inequality between the rank and the crank
of ordinary partitions. They obtained an asymptotic formula for the difference
between the rank and the crank of ordinary partitions, which implies
$N_S(m,n)\geq N_S(m+1,n)$ 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 by considering two cases. For $m\geq 1$, we
construct an injection from the set of ordinary partitions of $n$ such that $m$
appears in the rank-set to the set of ordinary partitions of $n$ with rank not
less than $-m$. The case for $m=0$ requires five more injections. We also show
that this conjecture implies an inequality between the positive rank and crank
moments obtained by Andrews, Chan and Kim.
