  Using the notion of Mahonian statistic on acyclic posets, we introduce a
$q$-analogue of the $h$-polynomial of a simple generalized permutohedron. We
focus primarily on the case of nestohedra and on explicit computations for many
interesting examples, such as $S_n$-invariant nestohedra, graph associahedra,
and Stanley--Pitman polytopes. For the usual (Stasheff) associahedron, our
generalization yields an alternative $q$-analogue to the well-studied Narayana
numbers.
