Abstract
In this article we describe a very efficient method to construct pairwise non-isomorphic posets (equivalently, T 0 topologies). We also give the results obtained by a computer program based on this algorithm, in particular the numbers of non-isomorphic posets on 15 and 16 points and the numbers of labelled posets and topologies on 17 and 18 points.
Similar content being viewed by others
References
Brinkmann, G. and McKay, B. D. Topologies and transitive relations up to 16 points, in preparation.
Chaunier, C. and Lyger?s, N. (1991) Progrés dans l'énumeration des posets, C.R. Acad. Sci. Paris Sér. I 314, 691–694.
Chaunier, C. and Lyger?s, N. (1992) The number of orders with thirteen elements, Order 9(3), 203–204.
Culberson, J. C. and Rawlins, G. J. E. (1991) New results from an algorithm for counting posets, Order 7(4), 361–374.
Das, S. K. (1977) A machine representation of finite T 0 topologies, J. ACM 24, 676–692.
El-Zahar, M. H. (1987) Enumeration of ordered sets, in J. Rival (ed.), Algorithms and Order, Kluwer Academic Publishers, Dordrecht, pp. 327–352.
Erné, M. and Stege, K. (1991) Counting finite posets and topologies, Order 8, 247–265.
Heitzig, J. and Reinhold, J. (2000) The number of unlabeled orders on fourteen elements, Order 17, 333–341.
Kleitman, D. J. and Rothschild, B. L. (1975) Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc. 205, 205–220.
Lyger?s, N. and Zimmermann, P. Computation of p(14); the number of posets with 14 elements: 1.338.193.159.771, http://www.desargues.univ-lyon1.fr/home/lygeros/poset.html.
McKay, B. D. (1981) Practical graph isomorphism, Congressus Numerantium 30, 45–87.
McKay, B. D. (1990) Nauty user's guide (version 1.5), Technical Report TR-CS-90-02, Australian National University, Department of Computer Science, http://cs.anu.edu.au/~bdm/nauty.
McKay, B. D. (1998) Isomorph-free exhaustive generation, J. Algorithms 26, 306–324.
Möhring, R. H. (1984) Algorithmic aspects of comparability graphs and interval graphs, in Graphs and Order: The Role of Graphs in the Theory of Ordered Sets and Its Applications, NATO Adv. Stud. Ser. C: Math. Phys. Sci. 147, pp. 41–102.
Prömel, H. J. (1987) Counting unlabeled structures, J. Combin. Theory, Ser. A 44, 83–93.
Wright, J. (1972) Cycle Indicators of Certain Classes of Types of Quasi-orders or Topologies, PhD Thesis, University of Rochester, NY.
Ziegler, J. F., Curtis, H. W., Muhlfeld, H. P., Montrose, C. J., Chin, B., Nicewicz, M., Russell, C. A., Wang, W. Y., Freeman, L. B., Hosier, P., LaFave, L. E., Walsh, J. L., Orro, J. M., Unger, G. J., Ross, J. M., O'Gorman, T. J., Messina, B., Sullivan, T. D., Sykes, A. J., Yourke, H., Enger, T. A., Tolat, V., Scott, T. S., Taber, A. H., Sussman, R. J., Klein, W. A. and Wahaus, C. W. (1996) IBM experiments in soft fails in computer electronics (1978-1994), IBM J. Res. Develop. 40(1), http://www.research.ibm.com/journal/rd/ziegl/ziegler.html.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brinkmann, G., McKay, B.D. Posets on up to 16 Points. Order 19, 147–179 (2002). https://doi.org/10.1023/A:1016543307592
Issue Date:
DOI: https://doi.org/10.1023/A:1016543307592