论文标题
排队夫妇的关键四重奏
A critical quartet for queuing couples
论文作者
论文摘要
我们列举了$ n $夫妇的安排，即成对的人，将单文件队列放置，并从杰出的给定夫妇的有利位置考虑四个统计数据。在$ n-1 $的其他夫妻中，$ n-1 $ p $ i）与给定夫妇交织在一起，ii）其中包含，iii）包含给定夫妇，而iv）躺在给定夫妇之外？我们提供生成的功能，以列举这些安排，并在$ n \ to \ infty $限制中获得相关的连续渐近分布。与情况相对应的渐近分布，iii）和iv）围绕$ p_c =（n-1）/2 $的临界现象，使得夫妇的可能性与其他一半以上的夫妻相结合，而2）夫妇在其他范围的一半中都包含了一半以上的$ ncere $ ncery $ n contty in y $ n c。我们进一步表明，在给定夫妇之外的其他夫妇中只有不到一半的累积概率是限制的$π/4 $，并且相关的分布对于$ p <p_c $均匀。
We enumerate arrangements of $n$ couples, i.e. pairs of people, placed in a single-file queue, and consider four statistics from the vantage point of a distinguished given couple. In how many arrangements are exactly $p$ of the $n-1$ other couples i) interlaced with the given couple, ii) contained within them, iii) containing the given couple, and iv) lying outside the given couple? We provide generating functions which enumerate these arrangements and obtain the associated continuous asymptotic distributions in the $n\to\infty$ limit. The asymptotic distributions corresponding to cases i), iii), and iv) evince critical phenomena around the value $p_c=(n-1)/2$, such that the probability that 1) the couple is interlaced with more than half of the other couples, and 2) the couple is contained by more than half of the other couples, are both zero in the strict $n\to\infty$ limit. We further show that the cumulative probability that less than half of the other couples lie outside the given couple is $π/4$ in the limit, and that the associated distribution is uniform for $p<p_c$.
