Orientations, Semiorders, Arrangements, and Parking Functions

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. In his study of Kazhdan-Lusztig cells of the affine Weyl group of typeAn−1, [10], J.-Y. Shi introduced the arrangement of hyperplanes in Rn now known as the Shi arrangement: xi − xj = 0, 1 1 ≤ i < j ≤ n. Among other things, he proved that the number of regions in the complement of this set of hyperplanes is (n+ 1)(n−1), Cayley’s formula for the number of trees on n+ 1 labeled vertices. The first bijective proof of this fact is due to Pak and Stanley, [11], who provide a method for labeling the regions with parking functions of size n. Given a graph G, Postnikov and Shapiro, [9], introduced the notion of a G-parking function. In the abelian sandpile model for G, [5], [2], these generalized parking functions are known as superstable configurations on G and are in bijection with the elements of the sandpile group for G. In the case where there exists a vertex q connected by edges to every other vertex of G, Duval, Klivans, and Martin, [6], have defined the G-Shi arrangement and conjecture that when its regions are labeled by the method of Pak and Stanley, the resulting labels are exactly the Gparking functions with respect to q. In this case, however, there may be duplicates among the labels. Letting G be the complete graph on n + 1 vertices recaptures the original result of Pak and Stanley. Our work was motivated by this conjecture. The semiorder arrangement, [12], is the set of n(n− 1) hyperplanes in Rn given by xi − xj = 1, i, j ∈ {1, . . . , n}, i 6= j. Its regions are in bijection with certain n-element posets called semiorders. In the same way that Duval, Klivans, and Martin modified Shi arrangements to take into account the structure of a graph, we modify semiorder arrangements to produce G-semiorder arrangements and label their regions using the method of Pak and Stanley. Theorem 25 shows that labels on a certain subset of the regions form the set of G-parking functions. This subset of