Generalized parking functions, descent numbers, and chain polytopes of ribbon posets

We consider the inversion enumerator In(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 In(−1)=En, 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.