New Wilf-equivalence results for vincular patterns

We give a sufficient condition for the two vincular patterns τ(1)−τ(2)−⋯−τ(ℓ) and τ(ℓ)−τ(ℓ−1)−⋯−τ(1) to be (strongly) Wilf-equivalent. This permits to solve in a unified way several problems of Heubach and Mansour on Wilf-equivalences on words and compositions, as well as a conjecture of Baxter and Pudwell on Wilf-equivalences on permutations. We also give a better explanation of the equidistribution of the parameters MAK+bMAJ and MAK′+bMAJ on ordered set partitions. Our results can be viewed as consequences of a proposition which states that the set valued statistics “descent set” and “rise set” are equidistributed over each equivalence class of the partially commutative monoid generated by a poset.