A critical quartet for queuing couples

TL;DR

This paper analyzes arrangements of couples in a queue, deriving generating functions and asymptotic distributions for various interlacing statistics, revealing critical phenomena and phase transitions as the number of couples grows large.

Contribution

It introduces a novel enumeration framework for couples in queues, providing generating functions and asymptotic analysis of interlacing statistics with critical phenomena insights.

Findings

Asymptotic distributions show phase transitions at p_c=(n-1)/2.

Probability of more than half couples interlaced or contained tends to zero as n→∞.

Distribution of outside couples is uniform below the critical point p_c.

Abstract

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 $\pi/4$ in the limit, and that the associated distribution is uniform for $p<p_c$.