Signs in the cd-index of Eulerian partially ordered sets

TL;DR

This paper characterizes the nonnegativity of the cd-index entries for all Eulerian posets, providing a complete description of which coefficients are nonnegative and establishing bounds on these coefficients.

Contribution

It completely determines the nonnegative entries of the cd-index for Eulerian posets and shows the absence of other bounds besides the trivial ones.

Findings

All entries of the cd-index are nonnegative for certain Eulerian posets.

No bounds other than trivial bounds exist for cd-coefficients.

The coefficient of c^n is the only universal bound.

Abstract

A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative cd-index. This paper completely determines which entries of the cd-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on cd-coefficients (except for the coefficient of c^n).