When you have a random list of n matching pairs of cards, what is the probability that a card and its match will be found side by side? Here is an example with n=3, where we take the card values (not suits) to define the matches. Here we have found the 3 beside its match.
Kreweras and Poupard answered this back in 1978. The answer is that the mean number of adjacent matches is 1, no matter the value of n. So if we shuffled the full deck, and defined a match by value and colour, we would have n=26, and would still expect just one adjacent match on average.
G. Kreweras and Y. Poupard, Sur les partitions en paires d’un ensemble fini totalement
ordonné, Publications de l’Institut de Statistique de l’Université de Paris 23 (1978),
57–74.
I've been having fun thinking about gadgets I call bubbles. Bubbles are sequences of cards delimited by these adjacent matches. For example below we have two bubbles, each of size two (the two sequences on either side of the paired 3's).
Here we have one bubble of size 4:
In my paper https://arxiv.org/abs/2311.01569 which appeared today I find a formal (non-convergent) generating function to count bubbles by their size. I also find that configurations consisting of a single bubble of size 2n are discontinuous in number compared to their smaller counterparts. Removing them, one obtains a smooth asymptotic distribution in the \(n \to \infty\) limit.
Nothing terribly deep or exciting for #combinatorics -- just very good recreational fun for me!