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arXiv · 1403.0303

Affine permutations and rational slope parking functions

Eugene Gorsky·Mikhail Mazin·Monica Vazirani
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Abstract

We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection exchanging the pairs of statistics (area,dinv) and (bounce,area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincare polynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials.

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Eugene Gorsky, Mikhail Mazin, Monica Vazirani. 2014-03-03. Affine permutations and rational slope parking functions. https://arxiv.org/abs/1403.0303

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