关于Smirnov词的下降和循环下降计数器
On enumerators of Smirnov words by descents and cyclic descents
摘要：
Smirnov字是一个在正整数上的字，其中相邻的字母必须不同。一个对称函数枚举这些词的下降数出现在工作Shareshian和第二个命名的作者对$q$ -欧拉多项式，其中一个$t$ -模拟公式的Carlitz，斯科维尔，沃恩枚举Smirnov的话是证明。一个对称函数列举了一个循环版本的这些话循环下降数出现在工作的第一个命名的作者色拟对称函数的有向图，其中一个$t$ -模拟的公式斯坦利列举循环Smirnov字证明。本文得到了Carlitz-Scoville-Vaughan公式和Stanley公式的新的$t$ -类似物，其中下降数和循环下降数的角色互换了.这些公式表明Smirnov字计数器是$t$中的多项式，其系数是e-正对称函数。我们还得到了幂和基和基本拟对称函数基的展开式，补充了Shareshian和作者的早期结果。我们的工作依赖于研究细化的Smirnov字计数器，计数某些限制类的Smirnov字的下降数。应用程序的变化$q$ -欧拉多项式和色拟对称功能介绍Shareshian和第二命名的作者。
英文摘要：
A Smirnov word is a word over the positive integers in which adjacent letters must be different. A symmetric function enumerating these words by descent number arose in the work of Shareshian and the second named author on $q$-Eulerian polynomials, where a $t$-analog of a formula of Carlitz, Scoville, and Vaughan for enumerating Smirnov words is proved. A symmetric function enumerating a circular version of these words by cyclic descent number arose in the work of the first named author on chromatic quasisymmetric functions of directed graphs, where a $t$-analog of a formula of Stanley for enumerating circular Smirnov words is proved. In this paper we obtain new $t$-analogs of the Carlitz-Scoville-Vaughan formula and the Stanley formula in which the roles of descent number and cyclic descent number are switched. These formulas show that the Smirnov word enumerators are polynomials in $t$ whose coefficients are e-positive symmetric functions. We also obtain expansions in the power sum basis and the fundamental quasisymmetric function basis, complementing earlier results of Shareshian and the authors. Our work relies on studying refinements of the Smirnov word enumerators that count certain restricted classes of Smirnov words by descent number. Applications to variations of $q$-Eulerian polynomials and to the chromatic quasisymmetric functions introduced by Shareshian and the second named author are also presented.
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