The r-signed Birkhoff transform

We introduce the r-signed Birkhoff transform of a distributive lattice, extending Hsiao's notion of the signed Birkhoff transform. We show how to compute the ab-index of the r-signed Birkhoff transform from the ab-index of the distributive lattice, generalizing work of Billera, Ehrenborg and Readdy, by extending their ω map to ω_r. We also obtain new expressions for the ab-index of the r-cubical lattice in terms of the map ω_r applied to all the permutations in the symmetric group. We show that the map r⋅ω_r=ϑ_r is an endomorphism on the Hopf algebra of quasisymmetric functions.