Some Wilf-equivalences for vincular patterns

Abstract

We extend the notion of shape-Wilf-equivalence to vincular (also known as generalized patterns or dashed patterns). First we introduce a stronger equivalence on which we call filling-shape-Wilf-equivalence. When vincular $\alpha$ and $\beta$ are filling-shape-Wilf-equivalent, we prove that the direct sum $\alpha\oplus\sigma$ is filling-shape-Wilf-equivalent to $\beta\oplus\sigma$. We also discover two new pairs of which are filling-shape-Wilf-equivalent: when $\alpha$, $\beta$, and $\sigma$ are nonempty consecutive which are Wilf-equivalent, $\alpha\oplus\sigma$ is filling-shape-Wilf-equivalent to $\beta\oplus\sigma$; and for any consecutive pattern $\alpha$, $1\oplus\alpha$ is filling-shape-Wilf-equivalent to $1\ominus\alpha$. These equivalences generalize Wilf-equivalences found by Elizalde and Kitaev. These new equivalences imply many new Wilf-equivalences for vincular