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تسجيل مستخدم جديدBlock decomposition and statistics arising from permutation tableaux
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تأليف
Joanna N. Chen
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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.
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