Abstract:We determine the exact thresholds for Schur positivity in the two families of chain products $\mathbf m\times\mathbf2$ and $\mathbf m\times\mathbf3$: the former is Schur positive exactly for $m\le7$, and the latter exactly for $m\le6$. For $m\ge8$, we prove non-Schur-positivity in both families by exhibiting explicit negative Schur coefficients obtained from Pieri's rules and stable-composition counts. The remaining finite cases are settled by exact SageMath computations; in particular, $\mathbf7\times\mathbf3$ has a negative Schur coefficient. These results settle the $n=2$ and $n=3$ cases in the conjectural picture of Li, Qiu, Yang, and Zhang and sharpen the $n=3$ boundary by one. We also show that $\mathbf m\times\mathbf3$ is not strongly nice for $m\ge44$.
From: Kai Zhang [view email]
[v1]
Fri, 3 Oct 2025 15:43:22 UTC (19 KB)
[v2]
Sun, 19 Jul 2026 08:50:49 UTC (70 KB)
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