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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Poset homology of Rees products, and q-Eulerian polynomials. | Poset homology of Rees products and q-Eulerian polynomials John Shareshian Department of Mathematics Washington University St. Louis MO 63130 shareshi@math.wustl.edu Michelle L. Wachs1 Department of Mathematics University of Miami Coral Gables FL 33124 wachs@math.miami.edu Submitted Oct 30 2008 Accepted Jul 24 2009 Published Jul 31 2009 Mathematics Subject Classifications 05A30 05E05 05E25 Dedicated to Anders Bjorner on the occasion of his 60th birthday Abstract The notion of Rees product of posets was introduced by Bjorner and Welker in 8 where they study connections between poset topology and commutative algebra. Bjorner and Welker conjectured and Jonsson 25 proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra Bn 0 and the n-chain Cn is equal to the number of derangements in the symmetric group n. Here we prove a refinement of this result which involves the Eulerian numbers and a q-analog of both the refinement and the original conjecture which comes from replacing the Boolean algebra by the lattice of subspaces of the n-dimensional vector space over the q element field and involves the maj exc -q-Eulerian polynomials studied in previous papers of the authors 32 33 . Equivariant versions of the refinement and the original conjecture are also proved as are type BC versions in the sense of Coxeter groups of the original conjecture and its q-analog. Supported in part by NSF Grants DMS 0300483 and DMS 0604233 and the Mittag-Leffler Institute Supported in part by NSF Grants DMS 0302310 and DMS 0604562 and the Mittag-Leffler Institute THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R20 1 Contents 1 Introduction and statement of main results 2 2 Preliminaries 6 3 Rees products with trees 10 4 The tree lemma 16 5 Corollaries 20 6 Type BC-analogs 22 1 Introduction and statement of main results In their study of connections between topology of order complexes and commutative algebra in 8 Bjorner and Welker introduced the .
