Proof of a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of M_{C1}(m,n) and M_{C5}(m,n)

Abstract

In their 2016 paper on exotic Bailey--Slater SPT-functions, Garvan and Jennings-Shaffer introduced many new spt-crank-type functions and proposed a conjecture that the spt-crank-type functions $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both nonnegative for all $m\in\mathbb{Z}$ and $n\in\mathbb{N}.$ Applying Wright\textquoteright s circle method, Jang and Kim showed that $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both positive for a fixed integer $m$ and large enough integers $n.$ Up to now, no complete proof of this conjecture has been given. In this paper, we provide a complete proof for this conjecture by using the theory of lattice points. Our proof is quite different from that of Jang and Kim.