More results on stack-sorting for set partitions

TL;DR

This paper investigates the properties of specific pattern-avoidance stack-sorting maps for sock sequences, providing algorithms to determine their images and analyzing the growth of preimages, advancing understanding of set partition sorting.

Contribution

It introduces two algorithms with $O(n^3)$ complexity for identifying sock sequences in the images of the maps and explores the exponential growth of preimages, deepening the theoretical understanding of these sorting maps.

Findings

Algorithms with $O(n^3)$ complexity for image determination.

Preimages grow at least exponentially in size.

Results on fertility numbers in set partition context.

Abstract

Let a sock be an element of an ordered finite alphabet A and a sequence of these elements be a sock sequence. In 2023, Xia introduced a deterministic version of Defant and Kravitz's stack-sorting map by defining the $\phi_{\sigma}$ and $\phi_{\overline{\sigma}}$ pattern-avoidance stack-sorting maps for sock sequences. Xia showed that the $\phi_{aba}$ map is the only one that eventually sorts all set partitions; in this paper, we prove deeper results regarding $\phi_{aba}$ and $\phi_{\overline{aba}}$ as a natural next step. We newly define two algorithms with time complexity $O(n^3)$ that determine if any given sock sequence is in the image of $\phi_{aba}$ or $\phi_{\overline{aba}}$ respectively. We also show that the maximum number of preimages that a sock sequence of length $n$ has grows at least exponentially under both the $\phi_{aba}$ and $\phi_{\overline{aba}}$ maps. Additionally, we prove results regarding fertility numbers (introduced by Defant) in the context of set partitions and multiple-pattern-avoiding stacks.