幂三角递推的多变量稳定性与参数化欧拉多项式

中文摘要 AI 辅助

我们证明了一类仿射三角递推的多变量稳定性，其中两个系数被提升到任意正整数次幂。这些变量在加权格点路径模型中跟踪跨步高度。应用包括参数化欧拉、斯特林和拉赫族的多变量稳定性与实根性。对于参数化欧拉多项式，我们获得了已知实根性结果的多变量细化，并证明了连续行的简单性和严格交错性。一个保持核的与递增二叉树的双射产生了对上的同步Foata--Strehl作用，以及它们的伽马系数的组合解释和递推关系。同一双射识别了允许对称布尔分解和链分解的细化偏序集（参数为1），以及覆盖支持的$\mathfrak{sl}_2$算子证明了它们的PECK性。我们将稳定性和严格交错结果扩展到同步斯特林排列，并证明了具有固定平台数的普通斯特林排列的PECK性。

英文摘要

We prove multivariate stability for a class of affine triangular recurrences whose two coefficients are raised to an arbitrary positive integer power. The variables keep track of cross-step heights in a weighted lattice-path model. The applications include stability and real rootedness of parametrised Eulerian, Stirling, and Lah families. For the parametrised Eulerian polynomials, we obtain multivariate refinements of known real-rootedness results and prove simplicity and strict interlacing of consecutive rows. A kernel-preserving bijection with increasing binary trees yields a synchronised Foata--Strehl action on pairs and a combinatorial interpretation of their gamma coefficients along with a recurrence. The same bijection identifies refinement posets that admit symmetric Boolean and chain decompositions for parameter one and cover-supported $\mathfrak{sl}_2$ operators prove their PECKness. We extend the stability and strict interlacing results to synchronised Stirling permutations and prove PECKness for ordinary Stirling permutations with a fixed number of plateaux.

发表机构

• Indian Institute of Science（印度科学研究所）

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