run-sorted排列上逆序统计量的分布

中文摘要 AI 辅助

设$\pi=\pi_1\cdots \pi_n$为一个排列。若$\pi_1=1$且$\pi$的下降位置之后紧接的元素构成递增序列，则称$\pi$为run-sorted排列。令$\mathcal{R}_n$表示长度为$n$的run-sorted排列的集合，其基数由贝尔数$B_{n-1}$给出（对所有$n \geq 1$）。本文研究$\mathcal{R}_n$上参数追踪逆序数和run数的联合分布$A_n(q,u)$，这导致贝尔数的一个新的多项式推广。在我们的结果中，我们找到了$A_n(q,u)$的一个一般递推关系，由此可以推导出$\mathcal{R}_n$中所有成员的逆序总数或run总数的显式公式，以及$\mathcal{R}_n$上任一参数的符号平衡。利用Gessel的$q$-指数公式，可以找到$A_n(q,1)$的欧拉生成函数的简单表达式，该公式可推广到一般的$u$。最后，通过直接论证找到了$\mathcal{R}_n$中成员内最大逆序数的公式。

英文摘要

Let $π=π_1\cdots π_n$ be a permutation. We say that $π$ is $run$-$sorted$ if $π_1=1$ and the entries immediately following the descent positions of $π$ form an increasing sequence. Let $\mathcal{R}_n$ denote the set of run-sorted permutations of length $n$, which has cardinality given by the Bell number $B_{n-1}$ for all $n \geq 1$. In this paper, we consider the joint distribution $A_n(q,u)$ on $\mathcal{R}_n$ for the parameters tracking the numbers of inversions and runs leading to a new polynomial generalization of the Bell numbers. Among our results, we find a general recurrence for $A_n(q,u)$, from which one may derive explicit formulas for the total numbers of inversions or runs in all the members of $\mathcal{R}_n$ as well as for the sign-balance on $\mathcal{R}_n$ of either parameter. A simple expression for the Eulerian generating function for $A_n(q,1)$ may be found upon making use of Gessel's $q$-exponential formula which can be extended to general $u$. Finally, a formula is found by a direct argument for the maximum number of inversions within a member of $\mathcal{R}_n$.

发表机构

• University of Haifa（海法大学）
• Makerere University（马凯雷雷大学）
• University of Tennessee（田纳西大学）

机构由 AI 辅助整理，请以论文原文为准。