Poset homology of Rees products and q-Eulerian polynomials

John Shareshian and Michelle L. Wachs

Abstract: The notion of Rees product of posets was introduced by Bj\"orner and Welker in \cite{bw}, where they study connections between poset topology and commutative algebra. Bj\"orner and Welker conjectured and Jonsson \cite{jo} 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 \cite{sw,ShWa}. 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.

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