Abstract
Pak and Stanley introduced a labeling of the regions of a k-Shi arrangement by k-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a graph G. They introduced the G-Shi arrangement and a labeling of its regions by G-parking functions. They conjectured that their labeling is surjective, i.e., that every G-parking function appears as a label of a region of the G-Shi arrangement. Later Hopkins and Perkinson proved this conjecture. In particular, this provided a new proof of the bijectivity of Pak-Stanley labeling in the k = 1 case. We generalize Hopkins-Perkinson’s construction to the case of arrangements associated with oriented multigraphs. In particular, our construction provides a simple straightforward proof of the bijectivity of the original Pak-Stanley labeling for arbitrary k.
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Athanasiadis C.A., Linusson S.: A simple bijection for the regions of the Shi arrangement of hyperplanes. Discrete Math. 204(1-3), 27–39 (1999)
Duval, A., Klivans, C., Martin, J.: The G-Shi arrangement, and its relation to G-parking functions. http://www.math.utep.edu/Faculty/duval/papers/nola.pdf (2011)
Foata D., Riordan J.: Mappings of acyclic and parking functions. Aequationes Math. 10, 10–22 (1974)
Gorsky E., Mazin M., Vazirani M.: Affine permutations and rational slope parking functions. Trans. Amer. Math. Soc. 368(12), 8403–8445 (2016)
Hopkins S., Perkinson D.: Bigraphical arrangement. Trans. Amer. Math. Soc. 368(1), 709–725 (2016)
Konheim A.G., Weiss B.: An occupancy discipline and applications. SIAM J. Appl. Math. 14(6), 1266–1274 (1966)
Postnikov A., Shapiro B.: Trees, parking functions, syzygies, and deformations of monomial ideals. Trans. Amer. Math. Soc. 356(8), 3109–3142 (2004)
Shi J.Y.: The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups. Springer-Verlag, Berlin (1986)
Stanley, R.P.: Hyperplane arrangements, parking functions and tree inversions. In: Sagan, B.E., Stanley, R.P. (eds.) Mathematical Essays in Honor of Gian-Carlo Rota. Progr. Math., Vol. 161, pp. 359–375. Birkhäuser Boston, Boston, MA (1998)
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Mazin, M. Multigraph Hyperplane Arrangements and Parking Functions. Ann. Comb. 21, 653–661 (2017). https://doi.org/10.1007/s00026-017-0368-7
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DOI: https://doi.org/10.1007/s00026-017-0368-7