Annals of Combinatorics

, Volume 10, Issue 2, pp 219–235 | Cite as

Arrangements and Ranking Patterns

  • Hidehiko Kamiya
  • Peter Orlik
  • Akimichi Takemura
  • Hiroaki Terao
Original Paper


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  ≤  8. We also consider the probability of the occurrence of each ranking pattern when the objects are randomly chosen.


unfolding model ranking pattern arrangement of hyperplanes characteristic polynomial mid-hyperplane arrangement spherical tetrahedron 

AMS Subject Classification.

32S22 52C35 62F07 


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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  • Hidehiko Kamiya
    • 1
  • Peter Orlik
    • 2
  • Akimichi Takemura
    • 3
  • Hiroaki Terao
    • 4
  1. 1.Faculty of EconomicsOkayama UniversityOkayamaJapan
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA
  3. 3.Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  4. 4.Department of MathematicsHokkaido UniversitySapporoJapan

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