Stack-sorting with Stacks Avoiding Vincular Patterns

We introduce the stack-sorting map ${\text{SC}}_{\sigma }$ that sorts, in a right-greedy manner, an input permutation through a stack that avoids some vincular pattern $\sigma$. The stack-sorting maps of Cerbai et al. in which the stack avoids a pattern classically and Defant and Zheng in which the stack avoids a pattern consecutively follow as special cases. We first characterize and enumerate the sorting class $\text{Sort}({\text{SC}}_{\sigma })$, the set of permutations sorted by $s\circ {\text{SC}}_{\sigma }$, for seven length $3$ patterns $\sigma$. We also decide when $\text{Sort}({\text{SC}}_{\sigma })$ is a permutation class. Next, we compute ${\max }_{\pi \in {\mathfrak{S}}_n}\mathrm{\mid }{\text{SC}}_{\sigma }^{-1}(\pi )\mathrm{\mid }$ and characterize the periodic points of ${\text{SC}}_{\sigma }$ for several length $3$ patterns $\sigma$. We end with several conjectures and open problems.