由排列、词和路径得到的实根Eulerian多项式
Real-rooted Eulerian polynomials from permutations, words, and paths
- Stockholm University（斯德哥尔摩大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究六个Eulerian型多项式族，证明错排下降多项式实根性等猜想，并加强或解决多个相关实根性与交错性问题。
AI中文摘要：
我们研究了六个Eulerian型多项式族。我们证明了错排的下降多项式是实根的，解决了S. Fu、Z. Lin和J. Zeng猜想中的错排部分。证明使用了相容对递归和有限符号稳定性。我们还解决了OEIS A335340中的实根性猜想，将偶数顶下降族的已知逐行实根性加强为连续严格交错，并证明了U. Shankar的超Eulerian多项式的实根性、连续交错性和实根gamma-多项式。一个微分递推给出了按递增游程计数的三元词的连续弱交错性。最后，我们证明了峰值细化及其正加权对角线的连续交错性的稳定性，解决了P. Alexandersson和O. Nabawanda的一个猜想。
英文摘要：
We study six Eulerian-type polynomial families. We prove that the descent polynomials of derangements are real-rooted, settling the derangement part of a conjecture of S.~Fu, Z.~Lin, and J.~Zeng. The proof uses a compatible-pair recursion and finite-symbol stability. We also resolve the real-rootedness conjecture in OEIS \oeis{A335340}, strengthen the known rowwise real-rootedness of an even-top descent family to consecutive strict interlacing, and prove real-rootedness, consecutive interlacing, and real-rooted gamma-polynomials for U.~Shankar's super-Eulerian polynomials. A differential recurrence gives consecutive weak interlacing for ternary words counted by increasing runs. Finally, we prove stability of the peak-value refinement and consecutive interleaving of its positive weighted diagonals, settling a conjecture of P.~Alexandersson and O.~Nabawanda.
