Recently Björner and Welker defined a poset product called the Rees product which is motivated by the Rees algebra of commutative algebra. They showed that the Rees product of two Cohen-Macaulay posets preserves the Cohen-Macaulay property. Very few examples of Rees products of Cohen-Macaulay posets have been studied. Jonsson showed the Möbius function of the Rees product of the Boolean algebra with the chain is a derangement number, solving an open problem of Björner and Welker. We show the Möbius function of the Rees product of the face lattice of the n-dimensional cube with the chain is given by n times a signed derangement number. One of the proofs we give is bijective between barred signed permutations which label the maximal chains of this poset and a set of labeled augmented skew diagrams. As a corollary, we give a bijective proof of Jonsson's result. We then use the labeled diagrams to index a basis for the homology of the order complex of the Rees product of the n-cube with the chain and determine a representation of the homology of its order complex over the symmetric group.
It is a classical result due to many authors, including Niven, de Bruijn and Viennot, that the alternating descent words abab...ab and baba...ba maximize the number of permutations having a particular descent pattern. Motivated by this work and the general program of determining all the flag vector inequalities among n-dimensional polytopes and more general manifolds, we study flag vector inequalities for the Dowling lattice, especially the special case of the partition lattice. Ehrenborg and Readdy conjectured that the maximum flag h-vector entry for the partition lattice occurs for the almost alternating descent words baba...bab or baba...babb. We show the flag vector of the partition lattice can be determined using a weighted boustrophedon transformation and determine many of the flag vector inequalities which hold among its flag vector entries. We end with a number of conjectures.