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We study the size of certain acyclic domains that arise from geometric andcombinatorial constructions. These acyclic domains consist of all permutationsvisited by commuting equivalence classes of maximal reduced decompositions ifwe consider the symmetric group and, more generally, of all c-singletons of aCambrian lattice associated to the weak order of a finite Coxeter group. Forthis reason, we call these sets Cambrian acyclic domains. Extending a closedformula of Galambos--Reiner for a particular acyclic domain called Fishburn'salternating scheme, we provide explicit formulae for the size of any Cambrianacyclic domain and characterize the Cambrian acyclic domains of minimum ormaximum size.

Cambrian acyclic domains: counting $c$-singletons