Proof of a conjecture of Garvan and Jennings-Shaffer on nonnegativity of <i>M</i> _{<i>C</i> 1} ( <i>m</i> , <i>n</i> ) and <i>M</i> _{<i>C</i> 5} ( <i>m</i> , <i>n</i> )

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’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 article, 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.