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Enumeration of Ordered Sets

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Algorithms and Order

Part of the book series: NATO ASI Series ((ASIC,volume 255))

Abstract

Given a class P of finite partially ordered sets (posets), then two counting problems are automatically raised. How many posets in P are there on n elements (enumeration of labelled posets) and what is the number of their isomorphism classes (enumeration of unlabelled posets)?

However, not all classes of posets are easy to enumerate. So, answers to these problems have different forms. If we have a sufficiently strong structure theorem, then we can manipulate the calculations in the algebra of formal power series. The required numbers are then obtained recursively from the resulting generating functions. Examples for this exact counting are labelled and unlabelled series-parallel posets [St], labelled and unlabelled interval orders [Ha], labelled and unlabelled graded posets [Kℓ1], [Kℓ2] and labelled tiered posets [Kr]. If we cannot carry out the exact counting then we content ourselves with an asymptotic estimate as in the case of unlabelled 2-dimensional posets [ES]. For other classes we have much cruder estimates. We have an asymptotic estimate for the logarithm of the number of labelled n-element posets [KR2]. For the logarithm of the number of n-element lattices we have a lower bound [KL] and an upper bound [KW] which differ by a multiplicative factor.

The purpose of this paper is to survey these results together with the techniques used in this area.

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References

  1. K.A. Baker, P.C. Fishburn and F.S. Roberts (1971) Partial orders of dimension 2, Networks 2, 11–28.

    Article  MathSciNet  Google Scholar 

  2. E.A. Bender (1974) Asymptotic methods in enumeration, SIAM Rev. 16, 485–515.

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Birkhoff (1967) Lattice Theory 3rd ed. Colloquium Publication 25, Amer. Math. Soc., Providence, R.I.

    MATH  Google Scholar 

  4. K.P. Bogart (1982) Some social science applications of ordered sets, in [Ri], 759–787.

    Google Scholar 

  5. N.G. de Bruijn, Polya theory of counting, in E.F. Beckenbach (ed.) (1964) Applied Combinatorial Mathematics Wiley, New York, 144–184.

    Google Scholar 

  6. B. Dushnik and E. Miller (1941) Partially ordered sets, Amer. J. Math. 63, 600–610.

    Article  MathSciNet  Google Scholar 

  7. M.H. El-Zahar and N. Sauer (1987) Asymptotic enumeration of two-dimensional posets, preprint.

    Google Scholar 

  8. P.C. Fishburn (1970) Intransitive indifference in preference theory: A survey, Oper. Res. 18, 207–228.

    Article  MathSciNet  MATH  Google Scholar 

  9. P.C. Gilmore and A.J. Hoffman (1964) A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16, 539–548.

    Article  MathSciNet  MATH  Google Scholar 

  10. M.C. Golumbic (1980) Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York.

    MATH  Google Scholar 

  11. P. Hanlon (1982) Counting interval graphs, Trans. Amer. Math. Soc. 272, 383–426.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Harary and E. Palmer (1973) Graphical Enumeration, Academic Press, New York.

    MATH  Google Scholar 

  13. D.A. Klarner (1969) The number of graded partially ordered sets, J. Combin. Theory 6, 12–19.

    Article  MathSciNet  MATH  Google Scholar 

  14. D.A. Klarner (1970) The number of classes of isomorphic graded partially ordered sets, J. Combin. Theory 9, 412–419.

    Article  MathSciNet  MATH  Google Scholar 

  15. D.A. Klarner (1986) The number of tiered posets modulo six, Discrete Math. 62, 295–297.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Kleitman and G. Markovsky (1975) On Dedekind’s problem: the number of isotone boolean functions. II, Trans. Amer. Math. Soc. 213, 373–390.

    MathSciNet  MATH  Google Scholar 

  17. D.J. Kleitman and B.L. Rothschild (1970) The number of finite topologies, Proc. Amer. Math. Soc. 25, 276–282.

    Article  MathSciNet  MATH  Google Scholar 

  18. D.J. Kleitjnan and B.L. Rothschild (1975) Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Hath. Soc. 205, 205–220.

    Article  Google Scholar 

  19. D.J. Kleitman and K.J. Winston (1980) The asymptotic number of lattices, Ann. Disc. Hath. 6, 243–249.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Klotz and L. Lucht (1971) Endliche Verbande, J. Reine Angew. Hath. 247, 58–68.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Korsunov (1977) Solution of Dedekind’s problem on the number of monotonic boolean functions, Dokl. Akad. Nauk SSR 223(4) 442–445

    MathSciNet  Google Scholar 

  22. A. Korsunov (1977) Solution of Dedekind’s problem on the number of monotonic boolean functions (Transi, by Amer. Math. Soc. in Soviet Hath. Dokl. 18(2), 442–445).

    MathSciNet  MATH  Google Scholar 

  23. G. Kreweras (1985) Denombrement des orderes etages, Discrete Hath 53, 147–149.

    Article  MathSciNet  MATH  Google Scholar 

  24. R.H. Mohring (1984) Almost all comparability graphs are UPO, Discrete Hath 50, 63–70.

    Article  MathSciNet  Google Scholar 

  25. H.J. Promel (1987) Counting unlabeled structures, J. Combin. Theor., Series A, 44, 83–93.

    Article  MathSciNet  Google Scholar 

  26. R.W. Quackenbush (1982) Enumeration in classes of ordered structures, in [Ri], 523–554.

    Google Scholar 

  27. I. Rival (ed.) (1982) Ordered Sets, Reidel, Dordrecht.

    MATH  Google Scholar 

  28. L.N. Shevrin and N.D. Filippov (1970) Partially ordered sets and their comparability graphs, Siber. Hath. J. 11, 497–509.

    Article  MATH  Google Scholar 

  29. R.P. Stanley (1974) Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Hath. Soc. 45(2), 295–299.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4

    Mohamed H. El-Zahar

  2. Department of Pure Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

    Mohamed H. El-Zahar

Authors
  1. Mohamed H. El-Zahar
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Editor information

Editors and Affiliations

  1. Departments of Computer Science and Mathematics, University of Ottawa, Ottawa, Canada

    Ivan Rival

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© 1989 Kluwer Academic Publishers

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El-Zahar, M.H. (1989). Enumeration of Ordered Sets. In: Rival, I. (eds) Algorithms and Order. NATO ASI Series, vol 255. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2639-4_9

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  • DOI: https://doi.org/10.1007/978-94-009-2639-4_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7691-3

  • Online ISBN: 978-94-009-2639-4

  • eBook Packages: Springer Book Archive

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