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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
Bigraphical Arrangements
We define the bigraphical arrangement of a graph and show that the
Pak-Stanley labels of its regions are the parking functions of a closely
related graph, thus proving conjectures of Duval, Klivans, and Martin and of
Hopkins and Perkinson. A consequence is a new proof of a bijection between
labeled graphs and regions of the Shi arrangement first given by Stanley. We
also give bounds on the number of regions of a bigraphical arrangement.Comment: Added Remark 19 addressing arbitrary G-parking functions; minor
revision
Posets, parking functions and the regions of the Shi arrangement revisited
The number of regions of the type A_{n-1} Shi arrangement in R^n is counted
by the intrinsically beautiful formula (n+1)^{n-1}. First proved by Shi, this
result motivated Pak and Stanley as well as Athanasiadis and Linusson to
provide bijective proofs. We give a description of the Athanasiadis-Linusson
bijection and generalize it to a bijection between the regions of the type C_n
Shi arrangement in R^n and sequences a_1a_2...a_n, where a_i \in \{-n,
-n+1,..., -1, 0, 1,..., n-1, n\}, i \in [n]. Our bijections naturally restrict
to bijections between regions of the arrangements with a certain number of
ceilings (or floors) and sequences with a given number of distinct elements. A
special family of posets, whose antichains encode the regions of the
arrangements, play a central role in our approach.Comment: 15 pages, 7 figure
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