Words Avoiding a Reflexive Acyclic Relation

Let A ⊆ [n] × [n] be a set of pairs containing the diagonal D = {(i, i) | i = 1, . . . , n}, and such that a ≤ b for all (a, b) ∈ A. We study formulae for the generating series FA(x) = ∑ w x w where the sum is over all words w ∈ [n]∗ that avoid A, i.e., (wi, wi+1) 6∈ A for i = 1, . . . , |w| − 1. This series is a rational function, with denominator of the form 1−∑T μA(T )xT , where the sum is over all nonempty subsets T of [n]. Our principal focus is the case where the relation A is μ-positive, i.e., μA(T ) ≥ 0 for all T ⊆ [n], in which case the form of the generating function suggests a cancellation-free combinatorial encoding of words avoiding A. We supply such an interpretation for several classes of examples, including the interesting class of cycle-free (or crown-free) posets.