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# Journal of Combinatorics

## Volume 2 (2011)

### Number 2

### The Rees product of posets

Pages: 165 – 191

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

#### Authors

#### 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

Published 25 October 2011