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

TL;DR

This paper explores generalized parking functions, their connection to descent sets in permutations, and provides combinatorial and geometric insights into their enumeration and associated polytopes.

Contribution

It extends classical parking function enumeration to generalized cases, linking them to descent sets and chain polytopes of ribbon posets with new combinatorial and geometric proofs.

Findings

Substituting q = -1 yields permutation counts with specific descent sets.

Volumes of generalized chain polytopes relate to sums over generalized parking functions.

Provides combinatorial and geometric proofs for these identities.

Abstract

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.