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The size of the maximum antichains in products of linear orders

Author

Listed:
  • Denis Bouyssou

    (LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

  • Thierry Marchant

    (UGENT - Universiteit Gent = Ghent University = Université de Gand)

  • Marc Pirlot

    (UMONS - Université de Mons / University of Mons)

Abstract

The size of maximum antichains in the product of n linear orders is known when the n linear orders have the same length. We present an exact expression for the size of maximum antichains when the linear orders have (possibly) different lengths. From this, we derive an exact expression for the size of maximum antichains in the product of n linear orders with the same length. This expression is equivalent to but different from the existing expression. It allows us to present an asymptotic result for the size of maximum antichains of n linear orders with the same length m going to infinity.

Suggested Citation

  • Denis Bouyssou & Thierry Marchant & Marc Pirlot, 2021. "The size of the maximum antichains in products of linear orders," Post-Print hal-03484128, HAL.
    • Handle: RePEc:hal:journl:hal-03484128
    DOI: 10.1007/s11750-020-00587-6
    Note: View the original document on HAL open archive server: https://hal.science/hal-03484128v1
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    References listed on IDEAS

    as
    1. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games," Documents de travail du Centre d'Economie de la Sorbonne 16081, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Hsiao, Chih-Ru & Raghavan, T E S, 1992. "Monotonicity and Dummy Free Property for Multi-choice Cooperative Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 301-312.
    3. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 301-326, July.
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    5. Josep Freixas, 2020. "The Banzhaf Value for Cooperative and Simple Multichoice Games," Group Decision and Negotiation, Springer, vol. 29(1), pages 61-74, February.
    6. Denis Bouyssou & Thierry Marchant & Marc Pirlot, 2020. "A theoretical look at ELECTRE TRI-nB," Working Papers hal-02917994, HAL.
    7. Michel Grabisch, 2016. "Rejoinder on: Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 335-337, July.
    8. Michel Grabisch, 2016. "Rejoinder on: Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 335-337, July.
    9. Mattner, Lutz & Roos, Bero, 2008. "Maximal probabilities of convolution powers of discrete uniform distributions," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2992-2996, December.
    10. Denis Bouyssou & Thierry Marchant & Marc Pirlot, 2020. "A theoretical look at ELECTRE TRI-nB," Working Papers hal-02898131, HAL.
    11. R. Branzei & N. Llorca & J. Sánchez-Soriano & S. Tijs, 2014. "A constrained egalitarian solution for convex multi-choice games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 860-874, October.
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    Cited by:

    1. Denis Bouyssou & Thierry Marchant & Marc Pirlot, 2023. "A theoretical look at Electre Tri-nB and related sorting models," 4OR, Springer, vol. 21(1), pages 1-31, March.

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    More about this item

    Keywords

    maximal antichain; Sperner; multichoice cooperative game; linear orders Mathematics Subject Classification (2010) 91B06;
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