Block decomposition and statistics arising from permutation tableaux.

Permutation statistics w m ¯ and rlm are both arising from permutation tableaux. w m ¯ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While rlm is shown 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 w m ¯ and rlm. Statistic (rlm, w m ¯ , rlmin, des, (321)) is shown equally distributed with (rlm, rlmin, w m ¯ , des, (321)) on S n. Then the generating function of (rlm, w m ¯) follows. An involution is constructed to explain the symmetric property of the generating function. Also, we study the triple statistic (w m ¯ , rlm, asc), which is shown to be equally distributed with (rlmax − 1, rlmin, asc) as studied by Josuat-Vergès. The main method we adopt throughout the paper is constructing bijections based on a block decomposition of permutations. [ABSTRACT FROM AUTHOR]