tailieunhanh - Báo cáo toán học: "Permutation Tableaux and the Dashed Permutation Pattern 32–1"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Permutation Tableaux and the Dashed Permutation Pattern 32–1. | Permutation Tableaux and the Dashed Permutation Pattern 32-1 William . Chen1 Lewis H. Liu2 1 2Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 . China 1chen@ 2lewis@ Submitted Dec 25 2010 Accepted May 4 2011 Published May 16 2011 Mathematics Subject Classifications 05A05 05A19 Abstract We give a solution to a problem posed by Corteel and Nadeau concerning permutation tableaux of length n and the number of occurrences of the dashed pattern 32-1 in permutations on n . We introduce the inversion number of a permutation tableau. For a permutation tableau T and the permutation n obtained from T by the bijection of Corteel and Nadeau we show that the inversion number of T equals the number of occurrences of the dashed pattern 32-1 in the reverse complement of n. We also show that permutation tableaux without inversions coincide with L-Bell tableaux introduced by Corteel and Nadeau. 1 Introduction Permutation tableaux were introduced by Steingrlmsson and Williams 14 in the study of totally positive Grassmannian cells 11 13 16 . They are closely related to the PASEP partially asymmetric exclusion process model in statistical physics 5 8 9 10 . Permutation tableaux are also in one-to-one correspondence with alternative tableaux introduced by Viennot 15 . A permutation tableau is defined by a Ferrers diagram possibly with empty rows such that the cells are filled with 0 s and 1 s subject to the following conditions 1 Each column contains at least one 1. 2 There does not exist a 0 with a 1 above in the same column and a 1 to the left in the same row . The length of a permutation tableau is defined as the number of rows plus the number of columns. A 0 in a permutation tableau is said to be restricted if there is a 1 above. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P111 1 Among the restricted 0 s in a row the rightmost 0 plays a special role which is called a rightmost restricted 0. A row is said to be unrestricted .

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