Consecutive patterns in Coxeter groups

Abstract

For an arbitrary Coxeter group element σ and a connected subset J of the Dynkin diagram, the parabolic decomposition σ=σJσJ defines σJ as a consecutive pattern of σ, generalizing the notion of consecutive patterns in permutations. We then define the cc-Wilf-equivalence classes as an extension of the c-Wilf-equivalence classes for permutations, and identify non-trivial families of cc-Wilf-equivalent classes. Furthermore, we study the structure of the consecutive pattern poset in Coxeter groups and prove that its Möbius function is bounded by 2 when the arguments belong to finite Coxeter groups, but can be arbitrarily large otherwise.