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.

Author Bio

I was born in Bogota, on July 21st, 1984. I started my undergraduate mathematics studies in 2001 at Universidad de los Andes, motivated by my participation in several national and international math olympiads while I was in highschool.During the time I spent at the university, I participated in the International Mathematics Competition for University Students obtaining good results. My fascination for mathematics increased every day, in several different areas such as combinatorics. This work on the Shi arrangement was made during the last year of my undergraduate studies, and was presented as part of my thesis, under the direction of Federico Ardila. I'm now finishing my master studies at the same university, and I hope to start my doctorate program in 2007 (if I can overcome the sadness of leaving Colombia).