Signs in the 𝑐𝑑-index of Eulerian partially ordered sets

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 c d cd -index. The c d 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 c d cd -index. This paper completely determines which entries of the c d cd -index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on c d cd -coefficients (except for the coefficient of c n c^n ).