Stack-Sorting with Dotted-Pattern-Avoiding Stacks

TL;DR

This paper introduces a new dotted pattern-avoiding map and extends the pattern-avoiding machine, providing analogs of classical results for these new maps and proposing several conjectures.

Contribution

It presents the first study of dotted pattern-avoiding maps and extends the pattern-avoiding machine to this context, establishing foundational results.

Findings

Analogous results for length 2 dotted pattern-avoiding maps

Extension of the pattern-avoiding machine to dotted patterns

Several conjectures proposed for future research

Abstract

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.