Báo cáo toán học: "Minimally Intersecting Set Partitions of Type B"

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: Minimally Intersecting Set Partitions of Type B. | Minimally Intersecting Set Partitions of Type B William . Chen and David . Wang Center for Combinatorics LPMC-TJKLC Nankai University Tianjin . China chen@ wgl@ Submitted Oct 6 2009 Accepted Jan 25 2010 Published Jan 31 2010 Mathematics Subject Classification 05A15 05A18 Abstract Motivated by Pittel s study of minimally intersecting set partitions we investigate minimally intersecting set partitions of type B. Our main result is a formula for the number of minimally intersecting r-tuples of Bn-partitions. As a consequence it implies the formula of Benoumhani for the Dowling number in analogy to Dobinski s formula. 1 Introduction This paper is primarily concerned with the meet structure of the lattice of type Bn partitions of the set 1 2 . n . The lattice of type Bn set partitions has been studied by Reiner 8 . It can be regarded as a representation of the intersection lattice of the type B Coxeter arrangements see Bjorner and Wachs 3 Bjorner and Brenti 2 and Humphreys 6 . A set partition of type Bn is a partition n of the set 1 2 . n into blocks satisfying the following conditions i For any block B of n its opposite B obtained by negating all elements of B is also a block of n ii There is at most one zero-block which is defined to be a block B such that B B. We call B a block pair of n if B is a non-zero-block of n. For example n 1 2 5 8 12 3 11 4 7 9 10 6 is a Bi2-partition consisting of 3 block pairs and the zero-block 1 2 5 8 12 . Our main result is a formula for the number of r-tuples of minimally intersecting Bn-partitions. We have used similar ideas in Pittel 7 but the variable setting for type B does not seem to be a straightforward generalization. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R22 1 Let us give a precise formulation of Pittel s results. Let nn be the lattice of partitions of n 1 2 . n . The minimum element in nn is 0 1 2 . n . The partitions n1 n2 . nr are said to intersect minimally if n1 A n2 A A nr

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