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arXiv · 0806.0440

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

Denis Chebikin·Alexander Postnikov
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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.

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Denis Chebikin, Alexander Postnikov. 2008-06-03. Generalized parking functions, descent numbers, and chain polytopes of ribbon posets. https://arxiv.org/abs/0806.0440

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