The r-cubical Lattice and a Generalization of the cd-index

In this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrér-signed permutations, a generalization of Purtillʹs work relating thecd-index of the cubical lattice and signed André permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Möbius function of arbitrary rank selections, where all theriʹs are greater than one, is the odd alternating ranks, {1, 3, 5, ...}.