The Basis of Foot-Sortable Sock Orderings

TL;DR

This paper characterizes the minimal initial sock arrangements that cannot be sorted using a specific stack-based process, providing a complete understanding of the problem's foundational configurations.

Contribution

It explicitly describes all minimal unsortable sock orderings under a defined stack operation model, advancing the theoretical understanding of sortable sequences.

Findings

Identified all minimal initial sock orderings that are unsortable.

Provided a complete characterization of sortable versus unsortable configurations.

Enhanced the theoretical framework for stack-based sorting problems.

Abstract

Defant and Kravitz considered the following problem: Suppose that, to the right of a foot, there is a line of colored socks that needs to be sorted. However, at any point in time, one can only either place the leftmost sock to the right of the foot onto the foot (stack) or remove the outermost sock on the foot and make it the rightmost sock to the left of the foot (unstack). In this paper, we explicitly describe all minimal initial sock orderings that are unsortable.