# 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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