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

In this paper we generalize the cd -index of the cubical lattice to an r cd -index , which we denote by Õ ( r ) . The coef ficients of Õ ( r ) enumerate augmented Andre ́ r -signed permutations , a generalization of Purtill’s work relating the cd -index of the cubical lattice and signed Andre ́ permutations . As an application we use the r cd -index to determine that the extremal configuration which maximizes the Mo ̈ bius function of arbitrary rank selections , where all the r i ’s are greater than one , is the odd alternating ranks , h 1 , 3 , 5 , . . . j .