Flag \(f\)-vectors and the \(cd\)-index
type: Item
instance of: scholarly article
title: Flag \(f\)-vectors and the \(cd\)-index
zbMATH Open document ID: 0805.06003
DOI: 10.1007/BF02572336
author: Richard P. Stanley
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174665
review text: 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.
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 704932
zbMATH Keywords: flag \(h\)-vector
zbMATH Keywords: Cohen-Macaulay poset
zbMATH Keywords: Gorenstein* poset
zbMATH Keywords: shelling
zbMATH Keywords: André polynomial
zbMATH Keywords: simplicial poset
zbMATH Keywords: flag \(f\)-vector
zbMATH Keywords: finite graded poset
zbMATH Keywords: number of chains
zbMATH Keywords: \(cd\)-index
zbMATH Keywords: face lattice of a convex polytope
zbMATH Keywords: Eulerian posets
zbMATH Keywords: Boolean algebras
zbMATH Keywords: noncommutative André permutations
MaRDI profile type: Publication
cites work: Posets, regular CW complexes and Bruhat order
cites work: A new index for polytopes
cites work: Shellable Decompositions of Cells and Spheres.
cites work: The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold
cites work: Q5603208
cites work: Which Spheres are Shellable?
cites work: Q3215224
cites work: Q5549086
cites work: Q1106472
cites work: Andre Permutations, Lexicographic Shellability and the cd-Index of a Convex Polytope
cites work: An unshellable triangulation of a tetrahedron
cites work: f-vectors and h-vectors of simplicial posets
OpenAlex ID: W2334336952
description: scientific article; zbMATH DE number 704932