1/k-欧拉多项式双γ正性的组合

IF 1.2 2区 数学 Q2 MATHEMATICS
Sherry H.F. Yan , Xubo Yang , Zhicong Lin
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引用次数: 0

摘要

1/k欧拉多项式An(k)(x)由Savage和Viswanathan作为k-反转序列上的上升多项式引入。1/k-欧拉多项式An(k)(x)的双γ正性是已知的,但给出相应的双γ系数的组合解释仍然是开放的。从纯组合的角度研究双γ正性的主题是由Athanasiadis提出的。本文利用有序标记森林模型,给出了An(k)(x)的双γ-系数的组合解释。我们的组合方法包括三个主要步骤:•在k-Stirling排列和某些被命名为增加修剪偶数k-ary森林的森林之间构造一个双射;•在增加修剪偶数k-ary树上引入一个广义的fota - strehl作用,该作用表明k-Stirling排列上的首字母为1的最长上升-高原多项式是γ-正的。•发展两个关键的转变,增加修剪甚至k-ary森林,以结束我们的组合解释。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Combinatorics on bi-γ-positivity of 1/k-Eulerian polynomials
The 1/k-Eulerian polynomials An(k)(x) were introduced as ascent polynomials over k-inversion sequences by Savage and Viswanathan. The bi-γ-positivity of the 1/k-Eulerian polynomials An(k)(x) was known but to give a combinatorial interpretation of the corresponding bi-γ-coefficients still remains open. The study of the theme of bi-γ-positivity from a purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-γ-coefficients of An(k)(x) by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps:
  • construct a bijection between k-Stirling permutations and certain forests that are named increasing pruned even k-ary forests;
  • introduce a generalized Foata–Strehl action on increasing pruned even k-ary trees which implies the longest ascent-plateau polynomials over k-Stirling permutations with initial letter 1 are γ-positive, a result that may have independent interest;
  • develop two crucial transformations on increasing pruned even k-ary forests to conclude our combinatorial interpretation.
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来源期刊
CiteScore
2.90
自引率
9.10%
发文量
94
审稿时长
12 months
期刊介绍: The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
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