We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we call these sets Cambrian acyclic domains. Extending a closed formula of Galambos--Reiner for a particular acyclic domain called Fishburn's alternating scheme, we provide explicit formulae for the size of any Cambrian acyclic domain and characterize the Cambrian acyclic domains of minimum or maximum size.

中文翻译：

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寒武纪非循环域：计数 c-singleton

我们研究由几何和组合结构产生的某些非循环域的大小。如果我们考虑对称群，并且更一般地，考虑与有限 Coxeter 群的弱阶相关联的寒武纪晶格的所有 c-singleton，则这些非循环域由最大简化分解的交换等价类访问的所有排列组成。为此，我们称这些集合为寒武纪非循环域。扩展 Galambos-Reiner 的封闭公式，用于称为 Fishburn 交替方案的特定无环域，我们提供了任何寒武纪无环域的大小的明确公式，并表征了最小或最大尺寸的寒武纪无环域。