Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bigraphical arrangements

Authors: Sam Hopkins and David Perkinson
Journal: Trans. Amer. Math. Soc. 368 (2016), 709-725
MSC (2010): Primary 52C35; Secondary 05C25
Published electronically: April 23, 2015
MathSciNet review: 3413881
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Drew Armstrong and Brendon Rhoades, The Shi arrangement and the Ish arrangement, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1509-1528. MR 2869184,
  • [2] Brian Benson, Deeparnab Chakrabarty, and Prasad Tetali, $ G$-parking functions, acyclic orientations and spanning trees, Discrete Math. 310 (2010), no. 8, 1340-1353. MR 2592488 (2011i:05152),
  • [3] Denis Chebikin and Pavlo Pylyavskyy, A family of bijections between $ G$-parking functions and spanning trees, J. Combin. Theory Ser. A 110 (2005), no. 1, 31-41. MR 2128964 (2005m:05010),
  • [4] Art Duval, Caroline Klivans, and Jeremy Martin, The $ G$-Shi arrangement, and its relation to $ G$-parking functions,, January 2011.
  • [5] Sam Hopkins and David Perkinson, Orientations, semiorders, arrangements, and parking functions, Electron. J. Combin. 19 (2012), no. 4, Paper 8, 31. MR 3001645
  • [6] Alexander Postnikov and Boris Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3109-3142 (electronic). MR 2052943 (2005a:05066),
  • [7] Jian Yi Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, vol. 1179, Springer-Verlag, Berlin, 1986. MR 835214 (87i:20074)
  • [8] Richard P. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), no. 6, 2620-2625. MR 1379568 (97i:52013),
  • [9] Richard P. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389-496. MR 2383131
  • [10] D. J. A. Welsh and C. Merino, The Potts model and the Tutte polynomial: Probabilistic techniques in equilibrium and nonequilibrium statistical physics, J. Math. Phys. 41 (2000), no. 3, 1127-1152. MR 1757953 (2001k:82026),
  • [11] Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. MR 0357135 (50 #9603)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 52C35, 05C25

Retrieve articles in all journals with MSC (2010): 52C35, 05C25

Additional Information

Sam Hopkins
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

David Perkinson
Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202

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

American Mathematical Society