Bell Permutation Tableaux

• Main Authors: Tze-chu Ho, 何澤初
• Other Authors: Sen-peng Eu
• Format: Others
• Language: en_US
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/kja3w5

碩士 === 國立高雄大學 === 應用數學系碩士班 === 100 === The concept of permutation tableaux was introduced by Postnikov in the context of enumeration of the totally positive Grassmannian cells. It is known that the number of permutation tableaux of length n is n!. Corteel and Nadeau gave a bijection Φ between the set of permutation tableaux and the set of permutations, and introduced L-Bell and R-Bell tableaux, both counted by the Bell numbers, as two subclasses of the permutation tableaux. Chen and Liu then characterized the corresponding permutations of the L-Bell tableaux under the bijection Φ. In this thesis we introduce three new subclasses of permutation tableaux, namely L'-Bell, R'-Bell, and B-Bell tableaux, and prove that they are also counted by the Bell numbers. We give characterizations of R-Bell and B-Bell tableaux in terms of pattern-avoiding permutations under the bijection Φ. We also introduce statistics on these five subclasses of tableaux and prove that they are equidistributed with the p,q-Stirling numbers of Wach and White on the 01-tableaux of Leroux. Meanwhile a new bijection consistent with the above q- statistic, but different from Corteel and Nadeau’s, is given between L-Bell tableaux and R-Bell tableaux. We also investigate the cardinality of the intersection of two of these subclasses. It turns out that many familiar classical sequences appear. Among them we prove that the cardinality of the intersection of the L-Bell and R-Bell tableaux (of the same length) is a Bessel number.