限制下降对 Quasi-Stirling 排列多项式的局部 γ-正性

Partial γ-Positivity for Quasi-Stirling Permutation with Restricted Descent Pair

摘要: 具有 γ - 正性的多项式在组合学中是一类重要的研究对象，其不仅蕴含了单峰性以及对称性，它的实根性在代数组合学中也有深远的研究意义。本文证明了仅包含 (偶，偶) - 下降对的 quasi- Stirling 排列上的一类三元多项式具有局部 γ - 正性，并给出局部 γ - 系数的组合解释，从而推广了 Eu 等人关于普通排列的相关结果。

Abstract: Polynomials with γ-positivity are an important research object in combinatorics. They not only contain unimodality and symmetry, but also have far-reaching research sig- nificance in algebraic combinatorics. This paper proves that a class of ternary polyno- mials that quasi-Stirling permutation only contains (even, even)-decreasing pairs has local γ-positivity, and the combined explanation of local γ-coefficient is given, thus extending the related results of Eu on ordinary permutation.