Consecutive patterns in inversion sequences II: avoiding patterns of relations

Inversion sequences are integer sequences e=e₁e₂ eₙ such that 0≤ eᵢ,=,≠\, we study inversion sequences e with no subindex i such that eᵢR₁eᵢ₊₁R₂eᵢ₊₂. By enumerating such inversion sequences according to their length, we obtain well-known quantities such as Catalan numbers, Fibonacci numbers and central polynomial numbers, relating inversion sequences to other combinatorial structures. We also classify consecutive patterns of relations into Wilf equivalence classes, according to the number of inversion sequences avoiding them, and into more restrictive classes that consider the positions of the occurrences of the patterns. As a byproduct of our techniques, we obtain a simple bijective proof of a result of Baxter--Shattuck and Kasraoui about Wilf-equivalence of vincular patterns, and we prove a conjecture of Martinez and Savage, as well as related enumeration formulas for inversion sequences satisfying certain unimodality conditions.