The Complete cd-Index of Boolean Lattices

Let [u, v] be a Bruhat interval of a Coxeter group such that the Bruhat graph BG(u, v) of [u, v] is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of [u, v]. Since in this case the complete cd-index and the cd-index of [u, v] coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in BG(u, v) by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in BG(u, v). Based on this labeling, we construct a flip F on the set of Bruhat paths in BG(u, v), which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip F is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial M . Thus by results of Karu, the coefficient of M enumerates certain Bruhat paths in BG(u, v), and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip F . We enumerate the number of cd-monomials in the complete cd-index of [u, v] in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.