2025/10/03 by David G. L. Wang, K. Zhang, Wang, David G. L. +1
Mathematics · #Advanced Combinatorial Mathematics #Random Matrices and Applications #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.2510.03116
We determine the exact thresholds for Schur positivity in the two families of chain products \mathbf m×\mathbf2 and \mathbf m×\mathbf3: the former is Schur positive exactly for m≤7, and the latter exactly for m≤6. For m≥8, 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×\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×\mathbf3 is not strongly nice for m≥44.