Affine permutations and rational slope parking functions

By E. Gorsky, Mikhail Mazin, Monica Vazirani. 2014

71 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/671f5ccf71fcd0a22c02e4ebf592ba60f29e3295

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In Algebra and Number Theory

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Citations in the corpus

Rational Parking Functions and Catalan Numbers (2014, background): “See [14, 15, 23] for related work that arises from a more geometric viewpoint.”