$F$- and $H$-Triangles for $\nu$-Associahedra and a Generalization of Klee's Dehn-Sommerville Relations.

For any northeast path $\nu$, we define two bivariate polynomials associated with the $\nu$-associahedron: the $F$- and the $H$-triangle. We prove combinatorially that we can obtain one from the other by an invertible transformation of variables. These polynomials generalize the classical $F$- and $H$-triangles of F. Chapoton in type $A$. Our proof is completely new and has the advantage of providing a combinatorial explanation of the relation between the $F$- and $H$-triangle. We also show that the specialization of this $F=H$ correspondence at $x=y$ is a special case of a general $f=h$ reciprocity result for abstract simplicial complexes, uncovering a clear explanation of the nature of this result. Interestingly, this connection leads to an elementary proof of Klee's Dehn-Sommerville relations, as well to a generalization in the context of simplicial homology manifolds and reciprocal complexes.