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- W1996588825 abstract "We give an interpretation for the coefficients of the two variable refinement D S n ( q , t ) of the distance enumerator of the Shi hyperplane arrangement S n 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." @default.
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- W1996588825 date "2008-07-01" @default.
- W1996588825 modified "2023-10-17" @default.
- W1996588825 title "On the two variable distance enumerator of the Shi hyperplane arrangement" @default.
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- W1996588825 doi "https://doi.org/10.1016/j.ejc.2007.08.002" @default.
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