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Bigraphical arrangements


Authors: Sam Hopkins and David Perkinson
Journal: Trans. Amer. Math. Soc. 368 (2016), 709-725
MSC (2010): Primary 52C35; Secondary 05C25
DOI: https://doi.org/10.1090/tran/6341
Published electronically: April 23, 2015
MathSciNet review: 3413881
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Abstract: 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 in 1996. We also give bounds on the number of regions of a bigraphical arrangement.


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Additional Information

Sam Hopkins
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: shopkins@mit.edu

David Perkinson
Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202
Email: davidp@reed.edu

DOI: https://doi.org/10.1090/tran/6341
Received by editor(s): December 23, 2012
Received by editor(s) in revised form: December 3, 2013
Published electronically: April 23, 2015
Article copyright: © Copyright 2015 American Mathematical Society