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Affine permutations and rational slope parking functions
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 ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the 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 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
Type C parking functions and a zeta map
We introduce type C parking functions, encoded as vertically labelled lattice
paths and endowed with a statistic dinv'. We define a bijection from type C
parking functions to regions of the Shi arrangement of type C, encoded as
diagonally labelled ballot paths and endowed with a natural statistic area'.
This bijection is a natural analogue of the zeta map of Haglund and Loehr and
maps dinv' to area'. We give three different descriptions of it.Comment: 12 page
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