Some combinatorial properties of the generalized derangement numbers

In this paper, we give a simple description of the m-widened permutations (generalized m-permutations) and the m-widened derangements (generalized m-derangements) in terms of ordinary permutations and derangements with a suitable constraint. This approach allows us to give a natural combinatorial interpretation of the generalized derangement numbers and the generalized rencontres polynomials in terms of species of structures. Finally, we obtain some formulas relating the generalized derangement numbers with the r-Bell numbers. In particular, we give an extension of the Clarke-Sued identity.