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.