Stack-Sorting with Dotted-Pattern-Avoiding Stacks

In this paper, we introduce the dotted pattern-avoiding map $s_{\dot{\tau}}$, which avoids the dotted pattern $\dot{\tau}$ instead of descents as West's stack-sorting map $s$ does. We also extend the pattern-avoiding machine, which is composed of a $\sigma$ avoiding map and West's stack-sorting map $s$, to the dotted pattern-avoiding machine. In this paper, we prove the analogs of the classical results on West's stack sorting map for the length $2$ dotted pattern-avoiding maps. We end with several conjectures.