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Mathematics and Computer Science III pp 3–14Cite as

Birkhäuser

Common Intervals of Permutations

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Part of the book series: Trends in Mathematics ((TM))

Abstract

Anintervalofapermutation isaconsecutive substring consisting of consecutive symbols. For example,4536 is an interval in the permutation 71453682. These arise in genetic applications. For the applications,it makes sense to generalise soas toallow gaps of bounded size S - 1,both in the locations andthesymbols. Forexample,4527 has gaps bounded by 1 (since3 and6 are missing) and is thereforeaS-interval of ****4*5*27**** for S =2.

After analysingthedistribution of the number of intervals ofauniform random permutation,we study the number of 2-intervals. This is exponentially large,but tightly clustered around itsmean.Perhaps surprisingly,the quenched and annealed means are the same. Our analysis isvia amultivariate generating function enumerating pairs of potential 2-intervals by size and intersection size.

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Author information

Authors and Affiliations

  1. CNRS PRiSM, Université de Versailles Saint-Quentin, 45 Avenue des Etats-Unis, Versailles, 78035, France

    Sylvie Corteel

  2. Département d’Informatique, Université Libre de Bruxelles, CP 212, Boulevard du Triomphe, Bruxelles, B-1050, Belgium

    Guy Louchard

  3. Department of Mathematics, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA, 19104, USA

    Robin Pemantle

Authors
  1. Sylvie Corteel
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  2. Guy Louchard
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  3. Robin Pemantle
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Editor information

Editors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10, Wien, 1040, Austria

    Michael Drmota  & Bernhard Gittenberger  & 

  2. INRIA Rocquencourt, Le Chesnay, 78153, France

    Philippe Flajolet

  3. PRISM, Université de Versailles-St-Quentin, Bâtiment Descartes 45 avenue des Etats-Unis, Versailles Cedex, 78035, France

    Danièle Gardy

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© 2004 Springer Basel AG

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Corteel, S., Louchard, G., Pemantle, R. (2004). Common Intervals of Permutations. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds) Mathematics and Computer Science III. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7915-6_1

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  • DOI: https://doi.org/10.1007/978-3-0348-7915-6_1

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9620-7

  • Online ISBN: 978-3-0348-7915-6

  • eBook Packages: Springer Book Archive

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