# Stack sorting with restricted stacks

## TL;DR

This paper studies a restricted stack sorting model called the σ-machine, where the first stack avoids a specific pattern, and analyzes when the set of sortable permutations forms a class, providing characterizations and enumerations for particular cases.

## Contribution

It introduces the σ-machine model with pattern-avoiding stacks and characterizes when the sortable permutations form a class, including detailed analysis for specific patterns.

## Key findings

- The set of σ-machines with non-class sortable permutations is counted by Catalan numbers.
- Complete characterization of sortable permutations for σ=321.
- Complete characterization of sortable permutations for σ=123.

## Abstract

The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be $\sigma$-avoiding, for some permutation $\sigma$, meaning that, at each step, the elements maintained in the stack avoid the pattern $\sigma$ when read from top to bottom. Since the set of permutations which can be sorted by such a device (which we call $\sigma$-machine) is not always a class, it would be interesting to understand when it happens. We will prove that the set of $\sigma$-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we will analyze two specific $\sigma$-machines in full details (namely when $\sigma=321$ and $\sigma=123$), providing for each of them a complete characterization and enumeration of sortable permutations.