The Rees product of posets

We 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 the permanent of a matrix. This gives 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 homology of the chain complex of the Rees product of the cubical lattice with the chain over the symmetric group. We also 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 is exactly the same as the Rees product of its dual with the chain.