Order

, Volume 9, Issue 4, pp 333–342 | Cite as

Random k-dimensional orders: Width and number of linear extensions

  • Graham Brightwell
Article
Received:
Accepted:

Abstract

We consider the width W k (n) and number L k (n) of linear extensions of a random k-dimensional order P k (n). We show that, for each fixed k, almost surely W k (n) lies between (√k/2−C)n1−1/k and 4kn1-1/k, for some constant C, and L k (n) lies between (e-2n1-1/k) n and (2kn1-1/k) n . The bounds given also apply to the expectations of the corresponding random variables. We also show that W k (n) and log L k (n) are sharply concentrated about their means.

Mathematics Subject Classifications (1991)

06A07 60C05 

Key words

Poset random order width linear extension 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Alon, B. Bollobás, G. Brightwell, and S. Janson: Linear extensions of a random partial order, to appear in Ann. Appl. Prob. Google Scholar
  2. 2.
    K. Azuma (1967) Weighted sums of certain dependent random variables, Tôhoku Math. Journal 19, 357–367.Google Scholar
  3. 3.
    B. Bollobás (1988) Martingales, isoperimetric inequalities and random graphs, in Combinatorics, A. Hajnal, L. Lovász and V. T. Sós (eds.), Colloq. Math. Sci. Janos Bolyai 52, North Holland.Google Scholar
  4. 4.
    B. Bollobás (1990) Sharp concentration of measure phenomena in random graphs, in Random Graphs '87, M. Karonski, J. Jaworski and A. Rucinski (eds.), John Wiley and Sons, New York, pp. 1–15.Google Scholar
  5. 5.
    B. Bollobás and G. Brightwell: The height of a random partial order: concentration of measure, to appear in Annals of Applied Prob. Google Scholar
  6. 6.
    B. Bollobás and G. Brightwell: Random high dimensional orders, submitted.Google Scholar
  7. 7.
    B. Bollobás and P. M. Winkler (1988) The longest chain among random points in Euclidean space, Proc. Amer. Math. Soc. 103, 347–353.Google Scholar
  8. 8.
    G. Brightwell: Linear extensions of random orders, to appear in Discrete Mathematics.Google Scholar
  9. 9.
    A. M. Frieze (1991) On the length of the longest monotone subsequence in a random permutation, Ann. Appl. Prob. 1, 301–305.Google Scholar
  10. 10.
    B. F. Logan and L. A. Shepp (1977) A variational problem for Young tableaux, Advances in Mathematics 26, 206–222.Google Scholar
  11. 11.
    C. J. H. McDiarmid (1989) On the method of bounded differences, in Surveys in Combinatorics 1989, Invited Papers at the 12th British Combinatorial Conference, J. Siemons (ed.), Cambridge University Press, 148–188.Google Scholar
  12. 12.
    S. Pilpel (1986) Descending subsequences of random permutations, IBM Research Report # 52283.Google Scholar
  13. 13.
    J. Spencer (1991) Nonconvergence in the theory of random orders, Order 7, 341–348.Google Scholar
  14. 14.
    A. M. Veršik and S. V. Kerov (1977) Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux, Dokl. Akad. Nauk. SSSR 233, 1024–1028.Google Scholar
  15. 15.
    P. M. Winkler (1985) Random orders, Order 1, 317–335.Google Scholar
  16. 16.
    P. M. Winkler (1985) Connectedness and diameter for random orders of fixed dimension, Order 2, 165–171.Google Scholar
  17. 17.
    P. M. Winkler (1989) A counterexample in the theory of random orders, Order 5, 363–368.Google Scholar
  18. 18.
    P. M. Winkler (1991) Random orders of dimension 2, Order 7, 329–339.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Graham Brightwell
    • 1
  1. 1.Department of Statistical and Mathematical SciencesLondon School of Economics and Political ScienceLondonUK

Personalised recommendations