Affine permutations and rational slope parking functions

Abstract

We introduce a new approach to the enumeration of rational slope parking functions with respect to the area \operatorname {area} and a generalized dinv \operatorname {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 ζ \zeta exchanging the pairs of statistics ( area , dinv ) (\operatorname {area},\operatorname {dinv}) and ( bounce , area ) (\operatorname {bounce}, \operatorname {area}) on Dyck paths, and the Pak-Stanley labeling of the regions of k k -Shi hyperplane arrangements by k 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 Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite-dimensional representations of DAHA and non-symmetric Macdonald polynomials.