Foot-Sorting for Socks

TL;DR

This paper introduces a novel foot-sorting algorithm for socks, characterizes sortable sock orderings, and establishes bounds on the number of feet needed for sorting based on sock variety.

Contribution

It presents a new foot-sorting algorithm, characterizes sortable sock arrangements, and proves bounds on the number of feet required for sorting socks of varying colors.

Findings

Characterization of sock orderings sortable with a fixed number of feet.

Enumeration of 1-foot-sortable sock orderings using Fibonacci numbers.

Maximum number of feet needed is logarithmic in the number of sock colors, and this bound is tight.

Abstract

If your socks come out of the laundry all mixed up, how should you sort them? We introduce and study a novel foot-sorting algorithm that uses feet to attempt to sort a sock ordering; one can view this algorithm as an analogue of Knuth's stack-sorting algorithm for set partitions. The sock orderings that can be sorted using a fixed number of feet are characterized by Klazar's notion of set partition pattern containment. We give an enumeration involving Fibonacci numbers for the $1$-foot-sortable sock orderings within a naturally-arising class. We also prove that if you have socks of $n$ different colors, then you can always sort them using at most $\left\lceil\log_2(n)\right\rceil$ feet, and we use a Ramsey-theoretic argument to show that this bound is tight.