发表机构
School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究链与极小偏序集乘积的反链多项式，给出计数公式、回文性条件，证明零点性质并构造双射，解决猜想并给出反例。
AI 中文摘要
本文研究了 $[k]\times P$ 的反链多项式，其中 $P$ 是连通的极小偏序集。我们给出了任意偏序集按大小计数的反链数量的公式。利用该公式，我们为两个无限族的连通极小偏序集给出了反链多项式回文性的充分必要条件。我们证明，对于每个连通的极小偏序集 $P$，若 $[k]\times P$ 的反链多项式是回文的，则它仅有实且严格为负的零点。这一结果特别对Ding-Dong关于$[k]\times P$的反链多项式的$\gamma$-正性猜想给出了肯定回答。通过构造反链与带标记的Dyck路径之间的双射，我们还给出了$[2]\times[m]\times[n]$中反链的层细化枚举。我们建立了这些反链与六边形薄片$O(2,m,n)$的Clar覆盖之间的关系，从而证明了一族Zhang-Zhang多项式的猜想行列式公式。当最短链长度至多为2时，还得到了实根性和稳定性结果。最后，我们给出了无限多个连通的Peck偏序集，其反链多项式不是单峰的，从而否证了Ding和Dong的对数凹性猜想。
英文摘要
This paper studies the antichain polynomials of $[k]\times P$, where $P$ is a connected minuscule poset. We give a formula for the number of antichains, counted by size, of an arbitrary poset. Using this formula, we present necessary and sufficient conditions for the palindromicity of antichain polynomials for two infinite families of connected minuscule posets. We show that, for every connected minuscule poset $P$, if the antichain polynomial of $[k]\times P$ is palindromic, then it has only real and strictly negative zeros. This result, in particular, gives an affirmative answer to Ding-Dong's conjecture about $γ$-positivity of the antichain polynomial of $[k]\times P$. By constructing a bijection between antichains and labeled Dyck paths, we also give a layer-refined enumeration of the antichains in $[2]\times[m]\times[n]$. We establish a relation between such antichains and Clar covers of the hexagonal flakes $O(2,m,n)$, thus prove a conjectured determinantal formula for a family of Zhang-Zhang polynomials. Real-rootedness and stability results are also obtained when the shortest chain has length at most two. Finally, we present infinitely many connected Peck posets whose antichain polynomials are not unimodal, disproving the log-concavity conjecture of Ding and Dong.
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