Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T06:30:16.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Counting of finite topologies and a dissection of Stirling numbers of the second kind

Published online by Cambridge University Press:  17 April 2009

V. Krishnamurthy
V. Krishnamurthy
Affiliation:
Department of Mathematics, Birla Institute of Technology and Science, Pilani (Rajasthan), India.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Certain new combinatorial numbers which arise in the counting of finite topologies are introduced and formulae obtained. These numbers are used to obtain a known formula for tn, the number of labelled topologies on n points in terms of the Stirling numbers S(n, p) and dn, the number of labelled T0-topologies on n points. The numbers dn are computed for n ≤ 5 with the the help of a method of Comtet (1966) (which seems to have been missed by later authors), reinterpreted for transitive digraphs.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1975 , pp. 111 - 124
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Louis, M. Comtet, “Recouvrements, bases de filtre et topologies d'un ensemble fini”, C.R. Acad. Sci. Paris Sér. A-B 262 (1966), A1091–A1094.Google Scholar
[2]Evans, J.W., Harary, F. and Lynn, M.S., “On the computer enumeration of finite topologies”, Comm. ACM 10 (1967), 295–297.CrossRefGoogle Scholar
[3]Gupta, Hansraj, “Number of topologies on a finite set”, Res. Bull. Panjab Univ. (U.S.) 19 (1968), 231–241.Google Scholar
[4]Krishnamurthy, V., “On the number of topologies on a finite set”, Amer. Math. Monthly 73 (1966), 154–157.CrossRefGoogle Scholar
[5]Rayburn, Marlon, “On the Borel fields of a finite set”, Proc. Amer. Math. Soc. 19 (1968), 885–889.CrossRefGoogle Scholar
[6]Shafaat, A., “On the number of topologies definable for a finite set”, J. Austral. Math. Soc. 8 (1968), 194–198.CrossRefGoogle Scholar
[7]Sharp, Henry Jr., “Quasi-orderings and topologies on finite sets”, Proc. Amer. Math. Soc. 17 (1966), 1344–1349.CrossRefGoogle Scholar