Journal of Combinatorics

Volume 2 (2011)

Number 2

The Rees product of posets

Pages: 165 – 191

DOI: http://dx.doi.org/10.4310/JOC.2011.v2.n2.a1

Authors

Patricia Muldoon Brown (Department of Mathematics, Armstrong Atlantic State University, Savannah, Ga., U.S.A.)

Margaret A. Readdy (Department of Mathematics, University of Kentucky, Lexington, Ky., U.S.A.)

Abstract

We determine how the flag $f$-vector of any graded poset changes under the Rees product with the chain, and more generally, any $t$-ary tree. As a corollary, the Möbius function of the Rees product of any graded poset with the chain, and more generally, the $t$-ary tree, is exactly the same as the Rees product of its dual with the chain, respectively, $t$-ary chain. We then study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give a bijective proof that the Möbius function of this poset can be expressed as $n$ times a signed derangement number. From this we derive a new bijective proof of Jonsson’s result that the Möbius function of the Rees product of the Boolean algebra with the chain is given by a derangement number. Using poset homology techniques we find an explicit basis for the reduced homology and determine a representation for the reduced homology of the order complex of the Rees product of the cubical lattice with the chain over the symmetric group.

Keywords

signed derangement numbers, poset products, flag vector, poset homology

2010 Mathematics Subject Classification

05A05, 05E10, 06A07

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