  We consider the inversion enumerator I_n(q), which counts labeled trees or,
equivalently, parking functions. This polynomial has a natural extension to
generalized parking functions. Substituting q = -1 into this generalized
polynomial produces the number of permutations with a certain descent set. In
the classical case, this result implies the formula I_n(-1) = E_n, the number
of alternating permutations. We give a combinatorial proof of these formulas
based on the involution principle. We also give a geometric interpretation of
these identities in terms of volumes of generalized chain polytopes of ribbon
posets. The volume of such a polytope is given by a sum over generalized
parking functions, which is similar to an expression for the volume of the
parking function polytope of Pitman and Stanley.
