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arXiv · 1009.4711

The Rees product of posets

Patricia Muldoon Brown·Margaret A. Readdy
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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 M\"obius 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 M\"obius 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 M\"obius 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.

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Patricia Muldoon Brown, Margaret A. Readdy. 2010-09-23. The Rees product of posets. https://doi.org/10.4310/joc.2011.v2.n2.a1

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