J an 2 00 1 Generalizations of Eulerian partially ordered sets , flag numbers , and the Möbius function

A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Möbius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets. Résumé Un ensemble partiellement ordonné est r-épais si chacun de ses intervals ouverts non-vides contient au moins r éléments. Dans cet This research was supported by University of Kansas General Research allocation #3552. On leave from the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences. Partially supported by Hungarian National Foundation for Scientific Research grant no. F 032325.