A brief overview of 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.