Cambrian acyclic domains: counting $c$-singletons

TL;DR

This paper provides explicit formulas for counting Cambrian acyclic domains, which are combinatorial structures related to Coxeter groups, extending previous work and characterizing extremal cases.

Contribution

It extends a known formula to all Cambrian acyclic domains and characterizes those with minimum or maximum size.

Findings

Derived explicit formulas for Cambrian acyclic domain sizes.

Characterized extremal Cambrian acyclic domains.

Extended previous combinatorial formulas to a broader class.

Abstract

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.