The Basis of Foot-Sortable Sock Orderings
Theodore Molla, Corey Nelson
TL;DR
This paper analyzes the problem of sorting a line of socks using two operations around a foot and provides an explicit description of the basis Γ of minimally unsortable sock orderings. The authors develop an algorithmic, greedy-based proof that any unsortable ordering must contain a subpattern from Γ, and they establish minimality of the Γ patterns via detailed case analyses and lemmas on sandwich structures. Central to the approach are the notions of states (S,R), sandwiches, and interlaced sequences I(T), which yield infinite families that underpin the basis. The work extends to discussions of sorting with multiple stacks and deque sorting, highlighting the rich structure of pattern avoidance in restricted-sorting models and connecting to prior results by Defant–Kravitz, Yu, and Xia, while suggesting directions for efficient decision procedures and further generalizations.
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.