A q-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets

Let f ( z ) = P ∞ n =0 ( − 1) n z n /n ! n !. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1 /f ( z ) = P ∞ n =0 ω n z n /n ! n !. They proved that ω n counts the number of pairs of permutations of the n th symmetric group S n with no common ascent. This paper gives a combinatorial interpretation of a natural q -analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n ( q ) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: P n i =0 ( − 1) i e i h n − i = 0, which then generalizes our q -analogue to a symmetric group representation result.