tailieunhanh - Báo cáo toán học: "The initial involution patterns of permutations"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The initial involution patterns of permutations. | The initial involution patterns of permutations Dongsu Kim Department of Mathematics Korea Advanced Institute of Science and Technology Daejeon 305-701 Korea dskim@ and Jang Soo Kim Department of Mathematics Korea Advanced Institute of Science and Technology Daejeon 305-701 Korea jskim@ Submitted July 18 2006 Accepted Dec 10 2006 Published Jan 3 2007 Mathematics Subject Classification 05A05 05A15 Abstract For a permutation 1 2 n 2 Sn and a positive integer i n we can view 1 2 i as an element of Si by order-preserving relabeling. The j-set of is the set of i s such that 1 2 i is an involution in Si. We prove a characterization theorem for j-sets give a generating function for the number of different j-sets of permutations in Sn. We also compute the numbers of permutations in Sn with a given j-set and prove some properties of them. 1 Introduction In order to count standard Young tableaux containing a given tableau McKay Morse and Wilf 2 considered the number of involutions in Sn containing a given permutation Ơ and Jaggard 1 found a formula for the number showing that the number depends only on the j-set of Ơ. The first author was partially supported by KRF grant R05-2004-000-11511-0. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R2 1 Let Sn denote the set of all permutations of n 1 2 . ng. A permutation V 2 Sn is called an involution if V V-1. Let w w1w2 Wj be a sequence of j distinct integers. The pattern of w is the permutation ơ ơ1 ơ2 ơj 2 Sj satisfying ơr ơs if and only if wr ws. For a permutation V V1V2 Vn 2 Sn the pattern of V1V2 V is called the initial i-pattern of V. Definition . The j-set of a permutation V 2 Sn with n 2 denoted by J is the set of all nonnegative integers i such that the initial i-pattern of V is an involution. For convenience we regard the initial 0-pattern as an involution. Jaggard in 1 Proposition uses j-sets to classify permutations according to subsequence containment by involutions. He gives data on