The Rees product of posets. (English) Zbl 1272.06004

Broen, Patricia Muldoon; Readdy, Margaret A.

J. Comb. 2, No. 2, 165-192 (2011).

Summary: 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

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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 cahin, 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

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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

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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.

MSC:

06A07	Combinatorics of partially ordered sets
05E10	Combinatorial aspects of representation theory

Keywords:

signed derangement numbers; poset products; flag vector; poset homology; poset topology; Rees product