Affine permutations and rational slope parking functions

Eugene Gorsky (1) , Mikhail Mazin (2) , Monica Vazirani (3)

1 Columbia University [New York]
2 Kansas State University
3 MATH - UC Davis - Department of Mathematics [Univ California Davis]

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 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.

Mots clés: Parking functions; affine permutations

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Eugene Gorsky, Mikhail Mazin, Monica Vazirani. Affine permutations and rational slope parking functions. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.887-898, ⟨10.46298/dmtcs.2450⟩. ⟨hal-01207556⟩