Block decomposition and statistics arising from permutation tableaux

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.

19pages,1 figure