Counting finite posets and topologies (Q1183945)

From authors' abstract: ``A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with \(n\) points and \(k\) antichains for \(n\) \(\leq 11\) and relevant integers \(k\). Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with \(n\) points for \(n\leq 14\). Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the number of finite topological spaces with \(n\) points and \(k\) open sets for \(n\leq 11\) and all \(k\), and then the numbers of all topologies on \(n\leq 14\) points satisfying various degrees on separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds \(10^{23}\).'' In addition to the new results cited above, the paper contains useful introductory and historical material on order relations and topologies.