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 and β⊕σ 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 is filling-shape-Wilf-equivalent to β⊕σ; and for any consecutive pattern α, 1⊕α is filling-shape-Wilf-equivalent to 1⊖α. These new equivalences imply many new Wilf-equivalences for vincular patterns.