A brief overview of the sock matching problem

TL;DR

This paper explores the sock matching problem by modeling sock draws as restricted Dyck paths, deriving formulas, and showing that the probability of encountering a specific number of unmatched socks approaches one as the total number of socks increases.

Contribution

It establishes new connections between the sock matching problem and Dyck paths, providing formulas and probabilistic results for large sock collections.

Findings

Probability of k unmatched socks approaches 1 as total socks grow large

Derived formulas linking sock matching to Dyck paths

Established new bounds and results for the sock matching problem

Abstract

This short note deals with the so-called $ Sock \; Matching \; Problem$. We define $B_{n,k}$ as the number of all the finite sequences $a_1, \ldots, a_{2n}$ of nonnegative integers which contain at least one occurrence of $k$ $(1 \leq k \leq n)$ and for which $a_1 = 1 $, $a_{2n}=0$ and $ \mid a_i -a_{i+1}\mid \; = 1$. The value $a_i$ can be interpreted as the number of unmatched socks being present after having drawn the first $i$ socks randomly out of the pile which initially contained $n$ pairs of socks. Here, establishing a link between this problem and with both some old and some new results, related to the number of restricted Dyck paths, we obtain a few valid forms of the sock matching theorem and prove that the probability for $k$ unmatched socks to appear (in the very process of drawing one sock at a time) approaches $1$ as the number of socks becomes large enough.