On the two variable distance enumerator of the Shi hyperplane arrangement

We give an interpretation for the coefficients of the two variable refinement DSn(q,t)DSn(q,t) of the distance enumerator of the Shi hyperplane arrangement SnSn in nn dimensions. This two variable refinement was defined by Stanley in [R.P. Stanley, Hyperplane arrangements, parking functions and tree inversions, in: B. Sagan, R. Stanley (Eds.), Mathematical Essays in Honor of Gian-Carlo Rota, Birkhauser, Boston, Basel, Berlin, 1998, pp. 359–375] for the general rr-extended Shi hyperplane arrangements.For the Shi hyperplane arrangement, we define three natural partitions of the number (n+1)n−1(n+1)n−1. The first arises from parking functions of length nn, the second from geometric considerations and the third from inversions on rooted spanning forests on nn vertices. We call the three partitions as the parking partition, the geometric partition and the inversion partition   respectively. We show that one of the parts of the parking partition is identical to the number of edge-labelled trees with label set {1,2,…,n}{1,2,…,n} on n+1n+1 unlabelled vertices. We prove that the parking partition majorises the geometric partition and conjecture that the inversion partition also majorises the geometric partition.