Counting Condorcet Domains

Abstract: A Condorcet domain is a collection of linear orders which satisfy an acyclic majority relation. In this paper we describe domains as collections of directed Hamilton paths. We prove that while Black's single-peaked domains are defined by their extremal paths, Arrow's single-peaked domains are not. We also introduce domain contractions and domain extensions as well as self-paired domains, and describe some properties of these. We give a formula for the number of isomorphism classes of Arrow's single-peaked domains in terms of the number of self-paired domains, and give upper and lower bounds on this number. We also enumerate the distinct maximal Arrow's single-peaked domains for $|A|=5,6,7, 8$. Finally, we show that all of the observations in this paper can be translated to single-dipped domains, that is, Condorcet domains with complete "never-top" conditions.