, Volume 9, Issue 1, pp 15–29 | Cite as

Enumeration of order preserving maps

  • Dwight Duffus
  • Vojtech Rodl
  • Bill Sands
  • Robert Woodrow


Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 22n/3 order preserving maps (and 22 in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.

Mathematics Subject Classification (1991)


Key words

(Partially) ordered set order preserving map enumeration stochastic process martingale 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Birkhoff (1967) Lattice Theory (3rd ed), AMS, Providence.Google Scholar
  2. 2.
    R. Canfield and Richard Duke (1991) private communication.Google Scholar
  3. 3.
    J. Currie and T. Visentin (1991) The number of order preserving maps of fences and crowns, Order 8, 133–142.Google Scholar
  4. 4.
    W. Hoefding (1963). Probability inequalities for sums of bounded random variables, J. Amer. Stat. Ass. 58, 13–30.Google Scholar
  5. 5.
    W.-P. Liu, I. Rival, and N. Zaguia (1991) Automorphisms, isotone self-maps and cycle-free orders, preprint.Google Scholar
  6. 6.
    I. Rival and A. Rutkowski (1990) Does almost every isotone self-map have a fixed point? preprint.Google Scholar
  7. 7.
    D. G. Robinson (1991) Fence endomorphisms and lattice paths with diagonal steps, preprint.Google Scholar
  8. 8.
    J. Spencer (1987) Ten Lectures on the Probabilistic Method, SIAM, Philadelphia.Google Scholar
  9. 9.
    R. G. Stanton and D. D. Cowan (1970) Note on a “square” functional equation. SIAM Review 12, 277–279.Google Scholar
  10. 10.
    W. T. Trotter (1989) private communication.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Dwight Duffus
    • 1
  • Vojtech Rodl
    • 1
  • Bill Sands
    • 2
  • Robert Woodrow
    • 2
  1. 1.Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.Mathematics and StatisticsThe University of CalgaryCalgaryCanada

Personalised recommendations