Given a graded poset, the authors define a so-called flag \(f\)-vector. Then the \(ab\)-index encodes that the flag \(f\)-vector of this poset as a non-commutative polynomial in the variables \(a\) and \(b\). Then they introduce the \(r\)-signed Birkhoff transform of a distributive lattice extending the known concept of the Birkhoff transform and show how to compute the \(ab\)-index of this \(r\)-signed Birkhoff transform from the \(ab\)-index of the distributive lattice. They obtain new expressions for the \(ab\)-index of the \(r\)-cubical lattice.