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Arrangements and Ranking Patterns
Hidehiko Kamiya1, Peter Orlik2, Akimichi Takemura3, and Hiroaki Terao4
1Faculty of Economics, Okayama University, Okayama 700-8530, Japan
kamiya@e.okayama-u.ac.jp
2Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
orlik@math.wisc.edu
3Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-0033, Japan
takemura@stat.t.u-tokyo.ac.jp
4Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
hterao@comp.metro-u.ac.jp
Annals of Combinatorics 10 (2) p. 219-235 June, 2006
AMS Subject Classification: 32S22, 52C35, 62F07
Abstract:
In the unidimensional unfolding model, given m objects in general position on the real line, there arise 1+m(m - 1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. Change of the position of these m objects results in change of the ranking pattern. In this paper we use arrangement theory to determine the number of ranking patterns theoretically for all m and numerically for m ¡Ü 8. We also consider the probability of the occurrence of each ranking pattern when the objects are randomly chosen.
Keywords: unfolding model, ranking pattern, arrangement of hyperplanes, characteristic polynomial, mid-hyperplane arrangement, spherical tetrahedron

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