scholarly journals Affine permutations and rational slope parking functions

2016 ◽  
Vol 368 (12) ◽  
pp. 8403-8445 ◽  
Author(s):  
Eugene Gorsky ◽  
Mikhail Mazin ◽  
Monica Vazirani
Keyword(s):  
Parking Functions ◽  
Affine Permutations

scholarly journals Affine permutations and rational slope parking functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Eugene Gorsky ◽  
Mikhail Mazin ◽  
Monica Vazirani
Keyword(s):  
Hyperplane Arrangements ◽  
Parking Functions ◽  
New Approach ◽  
Dyck Paths ◽  
International Audience ◽  
Affine Permutations ◽  
Combinatorial Constructions

International audience We introduce a new approach to the enumeration of rational slope parking functions with respect to the <mathrm>area</mathrm> and a generalized <mathrm>dinv</mathrm> 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 (<mathrm>area</mathrm>,<mathrm>dinv</mathrm>) and (<mathrm>bounce</mathrm>, <mathrm>area</mathrm>) 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.

scholarly journals 321-avoiding affine permutations and their many heaps

2019 ◽  
Vol 162 ◽  
pp. 271-305
Author(s):  
Riccardo Biagioli ◽  
Frédéric Jouhet ◽  
Philippe Nadeau
Keyword(s):  
Affine Permutations

scholarly journals Orientations, Semiorders, Arrangements, and Parking Functions

10.37236/2684 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Sam Hopkins ◽  
David Perkinson
Keyword(s):  
Hyperplane Arrangement ◽  
Parking Functions ◽  
Its Regions

It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph $G$, we define the $G$-semiorder arrangement and show that the Pak-Stanley labeling of its regions produces all $G$-parking functions.

scholarly journals A Decomposition of Parking Functions by Undesired Spaces

10.37236/5940 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Melody Bruce ◽  
Michael Dougherty ◽  
Max Hlavacek ◽  
Ryo Kudo ◽  
Ian Nicolas
Keyword(s):  
Parking Functions ◽  
Partition Lattice ◽  
Self Duality ◽  
Noncrossing Partition ◽  
Symmetric Chain Decomposition ◽  
Order Complexes ◽  
Maximal Chains ◽  
Symmetric Chain ◽  
Noncrossing Partition Lattice

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

scholarly journals Hopf algebras and dendriform structures arising from parking functions

2007 ◽  
Vol 193 (3) ◽  
pp. 189-241 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Jean-Yves Thibon
Keyword(s):  
Hopf Algebras ◽  
Parking Functions

scholarly journals Interval parking functions

2021 ◽  
Vol 123 ◽  
pp. 102129
Author(s):  
Emma Colaric ◽  
Ryan DeMuse ◽  
Jeremy L. Martin ◽  
Mei Yin
Keyword(s):  
Parking Functions
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