A generalization of the "probléme des rencontres"
In this paper, we study a generalization of the classical \emph{probl\'eme des rencontres} (\emph{problem of coincidences}),
consisting in the enumeration of all permutations $ \pi \in \SS_n $ with $k$ fixed points,
and, in particular, in the enumeration of all permutations $ \pi \in \SS_n $ with no fixed points (derangements).
Specifically, we study this problem for the permutations of the $n+m$
symbols $1$, $2$, \ldots, $n$, $v_1$, $v_2$, \ldots, $v_m$,
where $ v_i \not\in\{1,2,\ldots,n\} $ for every $i=1,2,\ldots,m$.