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

Abstract

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.