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Affine permutations and rational slope parking functions


Authors: Eugene Gorsky, Mikhail Mazin and Monica Vazirani
Journal: Trans. Amer. Math. Soc. 368 (2016), 8403-8445
MSC (2010): Primary 05E10, 05A05, 05A19, 20C08, 14M15
DOI: https://doi.org/10.1090/tran/6584
Published electronically: February 2, 2016
MathSciNet review: 3551576
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Abstract: We introduce a new approach to the enumeration of rational slope parking functions with respect to the $ \operatorname {area}$ and a generalized $ \operatorname {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 $ \zeta $ exchanging the pairs of statistics $ (\operatorname {area},\operatorname {dinv})$ and $ (\operatorname {bounce}, \operatorname {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.


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Additional Information

Eugene Gorsky
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
Address at time of publication: Department of Mathematics, University of California Davis, One Shields Avenue, Davis, California 95616 – and – National Research University – Higher School of Economics, Vavilova 7, Moscow, Russia
Email: egorsky@math.columbia.edu, egorskiy@math.ucdavis.edu

Mikhail Mazin
Affiliation: Department of Mathematics, Kansas State University, Cardwell Hall, Manhattan, Kansas 66506
Email: mmazin@math.ksu.edu

Monica Vazirani
Affiliation: Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, California 95616-8633
Email: vazirani@math.ucdavis.edu

DOI: https://doi.org/10.1090/tran/6584
Keywords: Parking functions, affine permutations
Received by editor(s): March 13, 2014
Received by editor(s) in revised form: October 4, 2014
Published electronically: February 2, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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