Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: SEuler characteristic of the truncated order complex of generalized noncrossing partitions. | Euler characteristic of the truncated order complex of generalized noncrossing partitions D. ÀRMSTRGNG AND C. KRATTENTHALERÍ Department of Mathematics University of Miami Coral Gables Florida 33146 USA. WWW http . edu armstrong Wakultat fur Mathematik Universitat Wien Nordbergstrabe 15 A-1090 Vienna Austria. WWW http .univie. ac . at kratt. Dedicated to Anders Bjorner on the occasion of his 60th birthday Submitted May 29 2009 Accepted Nov 23 2009 Published Nov 30 2009 2000 Mathematics Subject Classification Primary 05E15 Secondary 05A10 05A15 05A18 06A07 20F55 Abstract The purpose of this paper is to complete the study begun in the first author s PhD thesis of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups. 1 Introduction We say that a partition of the set n 1 2 . n is noncrossing if whenever we have a c in block A and b d in block B of the partition with a b c d it follows that Research partially supported by NSF grant DMS-0603567. Research partially supported by the Austrian Science Foundation FWF grants Z130-N13 and S9607-N13 the latter in the framework of the National Research Network Analytic Combinatorics and Probabilistic Number Theory. Key words and phrases. root systems reflection groups Coxeter groups generalized non-crossing partitions chain enumeration Euler characteristics Chu-Vandermonde summation. the electronic journal GF COMBINATORICS 16 2009 R143 1 A B. For an introduction to the rich history of this subject see 1 Chapter . We say that a noncrossing partition of mn is m-divisible if each of its blocks has cardinality divisible by m. The collection of m-divisible noncrossing partitions of mn which we will .
