On the two variable distance enumerator of the Shi hyperplane arrangement》.pdfVIP

European Journal of Combinatorics 29 (2008) 1104–1111 /locate/ejc On the two variable distance enumerator of the Shi hyperplane arrangement Sivaramakrishnan Sivasubramanian Institute of Information and Practical Mathematics, Christian-Albrechts-University, Kiel, Germany Received 5 January 2007; accepted 27 August 2007 Available online 22 October 2007 Abstract We give an interpretation for the coefficients of the two variable refinement DSn (q , t) of the distance enumerator of the Shi hyperplane arrangement Sn in n 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 r -extended Shi hyperplane arrangements. For the Shi hyperplane arrangement, we define three natural partitions of the number (n + 1)n −1. The first arises from parking functions of length n , the second from geometric considerations and the third from inversions on rooted spanning forests on n 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 } on n + 1 unlabelled vertices. We prove that the parking partition majorises the geometric partition and conjecture that the inversion partition also majorises the geometric partition. c 2007 Elsevier Ltd. All rights reserved. 1. Introduction Let r ≥ 1 and n ≥ 2. The