Proof of the Andrews-Dyson-Rhoades Conjecture on the spt-Crank

TL;DR

This paper proves the Andrews-Dyson-Rhoades conjecture that the distribution of the spt-crank of vector partitions is unimodal for all n, using a new representation called the m-Durfee rectangle symbol and combinatorial injections.

Contribution

The paper provides a proof of the conjecture by introducing the m-Durfee rectangle symbol and constructing combinatorial injections for different cases.

Findings

Confirmed the unimodality of the spt-crank distribution for all n.

Established a connection between the conjecture and inequalities of rank and crank moments.

Extended understanding of partition statistics through new combinatorial tools.

Abstract

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.