(M, i)-quasi-Stirling排列的欧拉多项式的实根性

Real-Rootedness of Eulerian Polynomials on (M, i)-quasi-StirlingPermutations

多重集上的 quasi-Stirling 排列作为 Stirling 排列的推广,其关于统计量的计数多项式的γ-正性、实根性等组合性质引起了众多学者的广泛关注。本文通过应用由Yan-Zhu 引入的quasi-Stirling排列与相关标号树之间的组合双射给出了(M, i)-quasi-Stirling 排列的欧拉多项式的递归关系,并在此基础上证明了该类多项式的实根性,从而得到了 Ma-Pan 关于 (M, i)-多重集排列的欧拉多项式实根性结论的类比结果。

quasi-Stirling permutations were introduced as a generalization of Stirling permuta- tions. The combinatorial properties of associated polynomials on quasi-Stirling permu- tations including the gamma-positivity and the real-rootedness have been extensively exploited in the literature. The main objective of this paper is to prove that theEulerian polynomial on (M, i)-quasi-Stirling permutations is real-rooted. This is ac-complished by deriving the recurrence relations on the related polynomials via the bijection between quasi-Stirling permutations and certain labeled trees introduced by Yan-Zhu. Our result is an analogue of the result due to Ma-Pan concerning thereal-rootedness of the Eulerian polynomial on (M, i)-permutations.