Fertility Numbers of Consecutive $S_3$ Pattern-Avoiding Stack-Sorting maps

TL;DR

This paper proves that for all length 3 patterns, every positive integer can be realized as a fertility number in a specific pattern-avoiding stack-sorting map, confirming a previous conjecture.

Contribution

It establishes that all positive integers are fertility numbers for all length 3 patterns in the specified stack-sorting maps, resolving a conjecture.

Findings

All positive integers are fertility numbers for the maps.

The result holds for all length 3 patterns.

The paper concludes with a new conjecture.

Abstract

In this paper, we show that for all length 3 patterns, all positive integers are fertility numbers for the consecutive-pattern-avoiding stack-sorting map $\textrm{SC}_\sigma$, which resolves conjecture 8.3 from Defant and Zheng. The paper ends with a conjecture.