Flag \(f\)-vectors and the \(cd\)-index
Tge flag \(f\)-vector \(\alpha_ P(S)\) of a finite graded poset \(P\) counts the number of chains \(C\) in \(P\) such that the ranks of the elements of \(C\) coincide with the elements of the set \(S\). If \(P\) is an Eulerian poset (e.g., the face lattice of a convex polytope), then the \(cd\)-index of \(P\) is a noncommutative polynomial \(\Phi_ P\) in the variables \(c\) and \(d\), first defined by J. Fine, which efficiently encodes the flag \(f\)-vector. Fine conjectured that when \(P\) is the face lattice of a convex polytope, \(\Phi_ P\) has nonnegative coefficients. We establish some new elementary properties of the \(cd\)-index and then prove Fine's conjecture for a class of posets more general than face lattices of polytopes. We conjecture that Fine's conjecture holds for all Cohen- Macaulay Eulerian posets and show that this would give all linear inequalities satisfied by the components of the flag \(f\)-vector of Cohen- Macaulay Eulerian posets. We specialize some of our results to Boolean algebras and obtain new information about the noncommutative André permutations of Foata and Schützenberger. Finally we obtain a formula for the \(cd\)-index of a simplicial Eulerian poset \(P\) which shows that the coefficients are nonnegative when \(P\) is also Cohen-Macaulay.
scientific article; zbMATH DE number 704932