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Báo cáo toán học: "The Skew and Relative Derangements of Type B"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Skew and Relative Derangements of Type B. | The Skew and Relative Derangements of Type B William Y.C. Chen1 and Jessica C.Y. Zhang2 Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 P. R. China 1chen@nankai.edu.cn 2zhangcy@mail.nankai.edu.cn Submitted Aug 27 2007 Accepted Oct 28 2007 Published Nov 5 2007 Mathematics Subject Classification 05E15 05A05 Abstract By introducing the notion of relative derangements of type B also called signed relative derangements which are defined in terms of signed permutations we obtain a type B analogue of the well-known relation between the relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements. 1 Introduction A derangement on a set n 1 2 ng is a permutation 1 2 n such that j i for all i 2 n . A relative derangement 1 2 n on n is a permutation such that i 1 i 1 for 1 i n 1. Let Qn denote the number of relative derangements on n and let Dn denote the number of the derangements on n . The following relation is well-known see Brualdi 2 Theorem 6.5.1 or Andreescu and Feng 1 Example 6.11 Qn Dn Dn 1 1.1 A combinatorial interpretation of 1.1 has been obtained by Chen 3 based on the intermediate structure of skew derangements which are equivalent to the generalized derangements as studied by Hanson Seyffarth and Weston 6 and Wang 8 . Clarke Han and Zeng 4 gave a similar construction based on variant of Foata s first fundamental transformation. The main objective of this paper is to present a type B analogue of THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N24 1 1.1 . This goal is achieved by introducing the notion of signed relative derangements or relative derangements of type B. The concept of derangements of type B is introduced by Chow 5 . A signed permutation K on n can be viewed as a bijection on the set 1 n 1 ng such that k z i . Intuitively a signed permutation on n