Ricerc@Sapienza

Let Dh, Ek and Falpha be sets of size h; k; alpha respectively, with k less than or equal to
h. A permutation of the union of Dh, Ek and Falpha such that the elements of Dh
are not fixed and the elements of Ek cannot occupy a site originally
occupied by an object of the same type or by an object of Falpha will be
called a strongly widened derangement. We show a connection between
strongly widened derangements and generalized Laguerre polynomials
that provides a generalization, for integer values of alpha, of Even and Gillis (1976) different

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 $i=1,2,\ldots,m$.