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

Abstract

Let f(z)=∑ 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)=∑ n=0 ∞ ω n z n /n!n!. They proved that ω n counts the number of pairs of permutations of the nth 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: ∑ i=0 n (-1) i e i h n-i =0, which then generalizes our q-analogue to a symmetric group representation result.