Theory of Probability &Its Applications, Volume 66, Issue 4, Page 708-712, February 2022.
Suppose that $n$ different pairs of socks are put in a tumble dryer. When the dryer is finished, socks are taken out one by one. If a sock matches one of the socks on the sorting table, both are removed; otherwise, it is put on the table until its partner emerges from the dryer. We note the number of socks on the table after each of the $2n$ socks is taken from the dryer, and we give an explicit formula for the probability that this sequence equals a given sequence of length $2n$.

中文翻译：

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袜子分类问题的路径公式

概率论及其应用，第 66 卷，第 4 期，第 708-712 页，2022 年 2 月。
假设将 $n$ 不同双袜子放在滚筒式烘干机中。当烘干机完成后，袜子会被一只一只地取出。如果一只袜子与排序表中的一只袜子匹配，则两者都被删除；否则，它会被放在桌子上，直到它的伙伴从烘干机中出来。我们注意到从烘干机中取出每只 $2n$ 袜子后桌子上的袜子数量，并且我们给出了该序列等于长度为 $2n$ 的给定序列的概率的显式公式。