Exact thresholds for Schur positivity of the lattices $\mathbf m\times\mathbf 2$ and $\mathbf m\times\mathbf 3$

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 $m\ge44$.