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A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the ag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of ag vectors of all graded posets. It also deÿnes a k-analogue of the M obius 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 ag vectors ofdoi:10.1016/s0012-365x(02)00336-9 fatcat:7s3kqgxum5bvbloipxnzntw7nq