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

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

2008-06-03

arxiv.org/abs/0806.0440v1

Mathematics

Combinatorics

11 pages, 1 figure

Scientific paper

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.

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