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

We determine sequences of polynomials with rational coefficients that have certain postulated values for their root power sums. These in turn determine four families of orthogonal polynomials that can be expressed in terms of Chebyshev polynomials by change of variable. We examine properties of these sequences of polynomials, two of which have already been studied in a previous paper.