Block decomposition and statistics arising from permutation tableaux. (January 2022)

• Record Type: Journal Article
• Authors: Chen, Joanna N.
• Is Part Of: European journal of combinatorics. Volume 99(2022)
• Publication Date: 2022-01
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103419
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: 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.