发表机构
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明三链乘积偏序集的分层细化反链多项式实稳定，其对角特化零点简单且严格为负，并解决相关猜想，强化伽马正性结论。
AI 中文摘要
对于所有正整数 $n,k$，我们证明了乘积偏序集 $[2]\ imes[n]\ imes[k]$ 的分层细化反链多项式是实稳定的。雅可比多项式交错进一步表明，其对角特化，即同一偏序集的普通反链多项式，仅具有单重且严格为负的零点。对于特殊族 $[2]\ imes[m]\ imes[m+1]$，显式的互反恒等式给出了回文性；简单负零点的互反对配对则表明伽马展开中的每个系数都严格为正。因此，我们证明了丁和董的猜想4.3，并解决了他们猜想4.5的所有部分，同时加强了其所述的伽马正性结论。计数输入是两个格路的显式首交叉反射，特化自克拉滕塔勒和苏兰克的工作。
英文摘要
For all positive integers $n,k$, we prove that the layer-refined antichain polynomial of the product poset $[2]\times[n]\times[k]$ is real stable. Jacobi-polynomial interlacing further shows that its diagonal specialization, the ordinary antichain polynomial of the same poset, has only simple, strictly negative zeros. For the special family $[2]\times[m]\times[m+1]$, explicit reciprocal identities give palindromicity; reciprocal pairing of the simple negative zeros then shows that every coefficient in the gamma expansion is strictly positive. Thus we prove Conjecture 4.3 of Ding and Dong and resolve all parts of their Conjecture 4.5, while strengthening its stated gamma-positivity consequence. The enumerative input is an explicit first-crossing reflection for two lattice paths, specialized from work of Krattenthaler and Sulanke.
Comments12 pages, with a 4-page ancillary computational supplement. Reproducibility package: https://doi.org/10.5281/zenodo.22082487
