Block decomposition and statistics arising from permutation tableaux

TL;DR

This paper explores the joint distribution of permutation statistics derived from permutation tableaux, introduces new bijections based on block decompositions, and analyzes their symmetric properties and generating functions.

Contribution

It introduces novel bijections and block decomposition techniques to study the joint distribution and symmetry of permutation statistics from permutation tableaux.

Findings

The joint distribution of multiple permutation statistics is characterized.

A generating function for the statistics is derived.

An involution explains the symmetry of the generating function.

Abstract

Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is showed by Nadeau equally distributed with the number of $1$'s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $\wnm$ and $\rlm$. Statistic $(\rlm,\wnm,\rlmin,\des,(\underline{321}))$ is shown equally distributed with $(\rlm,\rlmin,\wnm,\des,(\underline{321}))$ on $S_n$. Then the generating function of $(\rlm,\wnm)$ follows. An involution is constructed to explain the symmetric property of the generating function. Also, we study the triple statistic $(\wnm,\rlm,\asc)$, which is shown to be equally distributed with $(\rlmax-1,\rlmin,\asc)$ as studied by Josuat-Verg$\grave{e}$s. The main method we adopt throughout the paper is constructing bijections based on a block decomposition of permutations.