American Mathematical Society short course series: Fundamentals of applied combinatorics : August 12–13, 1977, University of Washington in Seattle,Seattle, Washington

A simple proof is given for the fact that the number of nonsingular similarity relations on {1, 2,… n}, for which the transitive closure of k blocks, equals $\text{(}{}^{\mathrm{2n?2k?1}}{}_{\mathrm{n?1}}\text{) ?(}{}^{\mathrm{2n?2k?1}}{}_{\mathrm{n}}\text{)}$1 ? k ? n > ?2. In particular, this implies a recent result of Shapiro about Catalan numbers and Fine's sequence.