Abstract
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. Nous introduisons les fonctions de stationnement de type $C$, encodées par des chemins étiquetés verticalement et munies d’une statistique dinv'. Nous définissons une bijection entre les fonctions de stationnement de type $C$ et les régions de l’arrangement de Shi de type $C$, encodées par des chemins étiquetés diagonalement et munies d’une statistique naturelle area'. Cette bijection est un analogue naturel à la fonction zeta de Haglund et Loehr, et envoie dinv' sur area'. Nous donnons trois différentes descriptions de celle-ci.
Highlights
Introduction and MotivationOne of the most well-studied objects in algebraic combinatorics is the space of diagonal harmonics of the symmetric group Sn
R∈Diagn where Parkn is the set of parking functions of length n, viewed as vertically labelled Dyck paths, and Diagn is the set of diagonally labelled Dyck paths with 2n steps
Let Dn denote the set of Dyck paths, that is the subset of Ln,n consisting of the paths that never go below the main diagonal x = y
Summary
The combinatorial objects Parkn and Diagn may be viewed as the type An−1 cases of more general objects associated to any crystallographic root system Φ These are, respectively, the finite torus Q/(h + 1)Qand the set of regions of the Shi arrangement of Φ. We introduce a statistic dinv’ on vertically labelled lattice paths and a statistic area’ on diagonally labelled ballot paths. These statistics are natural analogues of the corresponding statistics in type A. A full version of this extended abstract containing all proofs is in preparation
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