tailieunhanh - Báo cáo toán học: "Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns | Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns William Y. C. Chen1 Yu-Ping Deng2 and Laura L. M. Yang3 Center for Combinatorics LPMC Nankai University Tianjin 300071 P. R. China 1chenstation@ 2dengyp@ 3yanglm@ Submitted Mar 22 2003 Accepted Jun 18 2003 Published Aug 3 2003 MR Subject Classifications 05A05 05A15 Abstract We obtain a characterization of 321 3142 -avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of 321 3142 -avoiding permutations of length n equals the n-th Motzkin number due to Gire and further studied by Barcucci Del Lungo Pergola Pinzani and Guibert. Similarly we obtain a characterization of 231 4132 -avoiding permutations. For these two classes we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover we find a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path. 1. Introduction Permutations with forbidden subsequences have been extensively studied over the last decade. Simion and Schmidt 21 and West 26 initiated the efforts towards forbidden subsequences of length 3. West 25 and Stankova 22 continued to study forbidden subsequences of length 4 and discovered that the number of permutations in Sn 1 3142 2413 equals the n-th Schroder number. Using the idea of generating trees Kremer 17 discovered ten classes of permutations with forbidden patterns that are in one-to-one correspondence with Schrooder paths. Dyck paths are also closely related to permutations with forbidden patterns see Stanley 23 Krattenthaler 16 and West 26 . Motzkin paths come to the scene of permutations with forbidden patterns through the work of Gire 13 Barcucci Del Lungo Pergola and Pinzani 5 6 Guibert 14 and Guibert Pergola and Pinzani 15 . The classes Sn 321 3142 and Sn 231 4132