New Wilf-equivalence results for dashed patterns

Abstract

We give a sufficient condition for the two dashed patterns ${\tau }^{(1)}-{\tau }^{(2)}-\cdots -{\tau }^{(\ell )}$ and ${\tau }^{(\ell )}-{\tau }^{(\ell -1)}-\cdots -{\tau }^{(1)}$ 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. These results can be viewed as consequences of a simple 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 $(X,\le )$.