Poset Homology of Rees Products, and $q$-Eulerian Polynomials

The notion of Rees product of posets was introduced by Björner and Welker in [8], where they study connections between poset topology and commutative algebra. Björner and Welker conjectured and Jonsson [25] proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak{ S}$$_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the Boolean algebra by the lattice of subspaces of the $n$-dimensional vector space over the $q$ element field, and involves the (maj,exc)-$q$-Eulerian polynomials studied in previous papers of the authors [32,33]. Equivariant versions of the refinement and the original conjecture are also proved, as are type BC versions (in the sense of Coxeter groups) of the original conjecture and its $q$-analog.