Shape-Wilf-equivalences for vincular patterns

We extend the notion of shape-Wilf-equivalence to vincular patterns (also known as “generalized patterns” or “dashed patterns”). First we introduce a stronger equivalence on patterns which we call filling-shape-Wilf-equivalence. When vincular patterns α and β   are filling-shape-Wilf-equivalent, we prove that α⊕σ$\alpha \oplus \sigma$ and β⊕σ$\beta \oplus \sigma$ must also be filling-shape-Wilf-equivalent. We also discover two new pairs of patterns which are filling-shape-Wilf-equivalent: when α, β, and σ   are nonempty consecutive patterns which are Wilf-equivalent, α⊕σ$\alpha \oplus \sigma$ is filling-shape-Wilf-equivalent to β⊕σ$\beta \oplus \sigma$; and for any consecutive pattern α  , 1⊕α$1\oplus \alpha$ is filling-shape-Wilf-equivalent to 1?α$1?\alpha$. These new equivalences imply many new Wilf-equivalences for vincular patterns.