Set-alternating schemes: A new class of large Condorcet domains

In this paper, we introduce set-alternating schemes to construct Condorcet domains. These schemes lead to domains which are copious, connected, and peak-pit. The resulting family of domains includes some of Arrow's single-peaked domains of size $2^{n-1}$, which we prove to be the smallest possible domains. Still, schemes of this type lead to domains larger than the domains of Fishburn's alternating scheme. Thanks to the concise form of our schemes, we can analyze the growth of our fastest-growing domains. We show that the domain size for sufficiently high $n$ exceeds $2.1973^n$, improving the previous record $2.1890^n$ from \cite{karpov2023constructing}. To perform this analysis, a bijection between suborders and Dyck words, counted by the Catalan numbers, is developed.