A negative Schur coefficient for products of two chains

Abstract

We prove that the product of chains $\mathbf{m}\times\mathbf{n}$ is not Schur positive whenever $n\ge4$ and $m\ge3n-1$. Writing $m=n+k$, we exhibit a negative coefficient indexed by $(2n+k-1,2n+k-3,\ldots,k+5,k,k,4)$ and evaluate it explicitly as $(n-2)!$ times a polynomial of degree three in $k$. Our argument refines the chain-partition method of Li, Qiu, Yang, and Zhang: rank capacity forces $n-2$ long chains, and the remaining three chains are counted by separating a middle interval with fixed row coordinates from two bounded boundary regions. Together with their theorem and the known cases of widths two and three, this shows that these products are not Schur positive for $n\ge3$, $m\ge n+5$, and for $n=2$, $m\ge8$.