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Báo cáo toán học: " Mesh patterns and the expansion of permutation statistics as sums of permutation patterns"

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: Mesh patterns and the expansion of permutation statistics as sums of permutation patterns. | Mesh patterns and the expansion of permutation statistics as sums of permutation patterns Petter Branden Department of Mathematics Stockholm University SE-106 91 Stockholm Sweden Anders Claesson Department of Computer and Information Sciences University of Strathclyde Glasgow G1 1XH UK Submitted Feb 21 2011 Accepted Mar 3 2011 Published Mar 15 2011 Mathematics Subject Classification 05A05 05A15 05A19 Dedicated to Doron Zeilberger on the occasion of his sixtieth birthday Abstract Any permutation statistic f S C may be represented uniquely as a possibly infinite linear combination of classical permutation patterns f ETXf t t. To provide explicit expansions for certain statistics we introduce a new type of permutation patterns that we call mesh patterns. Intuitively an occurrence of the mesh pattern p n R is an occurrence of the permutation pattern n with additional restrictions specified by R on the relative position of the entries of the occurrence. We show that for any mesh pattern p n R we have Xp t 1 T - n p T where p n Rc is the mesh pattern with the same underlying permutation as p but with complementary restrictions. We use this result to expand some well known permutation statistics such as the number of left-to-right maxima descents excedances fixed points strong fixed points and the major index. We also show that alternating permutations Andre permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally we provide new natural Mahonian statistics. PB is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. 1AC was supported by grant no. 090038011 from the Icelandic Research Fund. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P5 1 1 Introduction 1.1 Mesh patterns Let a b be the integer interval i G Z a i b . Denote by n the set of permutations of 1 n . A mesh pattern is a pair p n R with n G k and R G 0 k X 0 k . An example .