Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2

Counting Transitive Relations

Götz Pfeiffer
Department of Mathematics
National University of Ireland, Galway
Ireland

Abstract: In order to count partial orders on a set of n points, it seems necessary to explicitly construct a representative of every isomorphism type. While that is done, one might as well determine their automorphism groups. In this note it is shown how several other types of binary relations can be counted, based on an explicit enumeration of the partial orders and their automorphism groups. A partial order is a transitive, reflexive, and antisymmetric binary relation. Here we determine the number of quasi-orders q(n) (or finite topologies or transitive digraphs or reflexive transitive relations), the number of "soft" orders s(t) (or antisymmetric transitive relations), and the number of transitive relations t(n) on n points in terms of numbers of partial orders with a given automorphism group.

(Concerned with sequences A000027 A000041 A000070 A000079 A000088 A000110 A000112 A000273 A000595 A000638 A000666 A000798 A001035 A001174 A001930 A002416 A006125 A006905 A047656 A053763 A079265 A083667 A083670 A091070 A091071 A091073 and A091566.)

Received January 22 2004; revised version received July 9 2004. Published in Journal of Integer Sequences August 2 2004.