袜子匹配问题的简要概述
A brief overview of the sock matching problem
摘要：
这篇短文涉及所谓的$ Sock \; Matching \; Problem$。我们定义$B_{n,k}$为所有非负整数的有限序列$a_1, \ldots, a_{2n}$的个数，这些序列至少包含一次$k$ $(1 \leq k \leq n)$，并且对于这些序列，$a_1 = 1 $，$a_{2n}=0$和$ \mid a_i -a_{i+1}\mid \; = 1$。值$a_i$可以解释为在从最初包含$n$双袜子的袜子堆中随机抽出第一双$i$袜子之后存在的不匹配袜子的数量。在这里，建立这个问题之间的联系，并与一些旧的和一些新的结果，有关的限制Dyck路径的数量，我们得到了一些有效的形式的袜子匹配定理，并证明了概率$k$不匹配的袜子出现（在一次绘制一只袜子的过程中）接近$1$的袜子的数量变得足够大。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
