Montmort matching problem

2020 Mathematics Subject Classification: Primary: 01A50 Secondary: 60-03 [MSN][ZBL]

derangement problem, problème des rencontres

Take two sets, $A$ and $B$, and a bijection, $\phi$, between them. (E.g., take $n$ married couples and let $A$ be the set of husbands and $B$ the set of wives.) Now, take a random pairing (a bijection again). What is the chance that this random pairing gives at least one "correct match" (i.e. coincides with $\phi$ in at least one element). Asymptotically, this probability is $1-e^{-1}$. This follows immediately from the formula given in Classical combinatorial problems for the number of derangements: permutations $\pi$ such that $\pi(i)\neq i$ for all $i=1,\ldots,n$.

This problem was considered first by P.R. de Montmort (around 1700) in connection with a card game known as the "jeu du treize", "jeu de rencontre" or simply "rencontre".

References

[a1]	E. Knobloch, "Euler and the history of a problem in probability theory" Ganita–Bharati , 6 (1984) pp. 1–12