Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Wreath Products of Permutation Classes. | Wreath Products of Permutation Classes Robert Brignall School of Mathematics and Statistics University of St Andrews St Andrews Fife Scotland robertb@ http robertb Submitted Sep 28 2006 Accepted Jun 3 2007 Published Jun 28 2007 Mathematics Subject Classification 05A05 06A07 Abstract A permutation class which is closed under pattern involvement may be described in terms of its basis. The wreath product construction X o Y of two permutation classes X and Y is also closed and we exhibit a family of classes Y with the property that for any finitely based class X the wreath product X o Y is also finitely based. Additionally we indicate a general construction for basis elements in the case where X o Y is not finitely based. 1 Introduction and Statement of Theorem Two finite sequences of the same length a a1a2 an and b b1b2 bn are said to be order isomorphic if for all i j we have ai Oj if and only if bi bj. Viewing permutations of length n as orderings on the numbers 1 2 . n every sequence of n distinct symbols is order isomorphic to a unique permutation. A permutation Ơ is said to be involved in the permutation denoted Ơ if there is a subsequence or pattern of order isomorphic to Ơ. For example 1324 6351427 because of the subsequence 3547. A book introducing the study of these permutation patterns has been written by Bona 6 . This involvement order forms a partial order on the set of all finite permutations sets of permutations which are closed downwards under this order are called permutation classes. These classes are specified primarily in one of three ways Pattern avoidance. A permutation class X can be regarded as a set of permutations which avoid certain patterns. The set B of minimal permutations not in X yFor a sequence a not necessarily a permutation and set of permutations Y with a slight abuse of notation we will sometimes write statements like a 2 Y meaning the permutation order isomorphic to a lies in Y. THE .
