The complete \(\mathbf{cd}\)-index of Boolean lattices
Summary: Let \([u,v]\) be a Bruhat interval of a Coxeter group such that the Bruhat graph \(\mathrm{BG}(u,v)\) of \([u,v]\) is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete \(\mathbf{cd}\)-index of \([u,v]\). Since in this case the complete \(\mathbf{cd}\)-index and the \(\mathbf{cd}\)-index of \([u,v]\) coincide, we also obtain a new combinatorial interpretation for the coefficients of the \(\mathbf{cd}\)-index of Boolean lattices. To this end, we label an edge in \(\mathrm{BG}(u,v)\) 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 \(\mathrm{BG}(u,v)\). Based on this labeling, we construct a flip \(\mathcal{F}\) on the set of Bruhat paths in \(\mathrm{BG}(u,v)\), which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip \(\mathcal{F}\) is compatible with any given reflection order and also satisfies the flip condition for any \(\mathbf{cd}\)-monomial \(M\). Thus by results of Karu, the coefficient of \(M\) enumerates certain Bruhat paths in \(\mathrm{BG}(u,v)\), and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip \(\mathcal{F}\). We enumerate the number of \(\mathbf{cd}\)-monomials in the complete \(\mathbf{cd}\)-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 \(\mathbf d^n\) in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.
scientific article; zbMATH DE number 6475719
scientific article