Boolean intervals in the weak Bruhat order of a finite Coxeter group

Given a Coxeter group W with Coxeter system (W, S), where S is finite. We provide a complete characterization of Boolean intervals in the weak order of W uniformly for all Coxeter groups in terms of independent sets of the Coxeter graph. Moreover, we establish that the number of Boolean intervals of rank k in the weak order of W is iₖ (W) |W|\, /\, 2^k, where W is the Coxeter graph of W and iₖ (W) is the number of independent sets of size k of W when W is finite. Specializing to Aₙ, we recover the characterizations and enumerations of Boolean intervals in the weak order of Aₙ given in arXiv: 2306. 14734. We provide the analogous results for types Cₙ and Dₙ, including the related generating functions and additional connections to well-known integer sequences.