Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A generalization of Simion-Schmidt’s bijection for restricted permutations | A generalization of Simion-Schmidt s bijection for restricted permutations Astrid Reifegerste Institut fur Mathematik Universitat Hannover Welfengarten 1 D-30167 Hannover Germany reifegerste@ Submitted Mar 27 2003 Accepted Jun 9 2003 Published Jun 18 2003 MR Subject Classifications 05A05 05A15 Abstract. We consider the two permutation statistics which count the distinct pairs obtained from the final two terms of occurrences of patterns T1 Tm-2m m 1 and T1 Tm-2 m 1 m in a permutation respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As a special case we derive a one-to-one correspondence between permutations which avoid each of the patterns T1 Tm-2m m 1 G Sm and those which avoid each of the patterns T1 Tm-2 m 1 m G Sm. For m 3 this correspondence coincides with the bijection given by Simion and Schmidt in 11 . 1 Introduction Recently a lot of work has been done investigating permutations with restrictions on the patterns they contain. Given a permutation n G Sn and a permutation T G Sm an occurrence of T in n is an integer sequence 1 ii i2 . im n such that the letters of the subword nir ni2 nim are in the same relative order as the letters of T. In this context T is called a pattern. If there is no occurrence at all we say that n avoids T or alternatively n is T-avoiding. We write Sn T to denote the set of T-avoiding permutations in Sn and more generally Sn T for the set of all permutations of length n which avoid each pattern of the set T. A central theme in the theory of pattern-avoiding permutations is to classify all patterns up to Wilf-equivalence. Two patterns T1 and T2 are called Wilf-equivalent if they are equally restrictive that is Sn ti S t2 for all n G N. The first major result dealing with this problem states that 123 and 132 are Wilf-equivalent. By obvious symmetry arguments this implies that S3 is one Wilf-class. The first explicit bijection between
