Let Fn be the face poset of the n-dimensional Shi arrangement, and let Pn be the poset of parking functions of length n with the order defined by (a1, a2 , , an ) <= (b1 , b2 , , bn ) if ai <= bi for all i. Pak and Stanley constructed a labelling of the regions in Fn using the elements of Pn. We show that under this labelling, all faces in Fn correspond naturally to closed intervals of Pn, so the labelling of the regions can be extended in a natural way to a labelling of all faces in Fn. We also explore some interesting and unexpected properties of this bijection. We finally give some results that help to characterize the intervals that appear as labels and consequently to obtain a better comprehension of Fn. As an application we are able to count in a bijective way the number of one dimensional faces.
"A Labelling of the Faces in the Shi Arrangement,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 8:
1, Article 7.
Available at: https://scholar.rose-hulman.edu/rhumj/vol8/iss1/7