Non-Eulerian Dehn-Sommerville relations

The classical Dehn--Sommerville relations assert that the \(h\)-vector of an Eulerian simplicial complex is symmetric. We establish three generalizations of the Dehn--Sommerville relations: one for the \(h\)-vectors of pure simplicial complexes, another one for the flag \(h\)-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric \(h\)-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of "errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relations.