  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 $p<p_c$.
