Combinatorial interpretations and enumeration of particular bijections

Abstract. Let n be a nonnegative integer. We call widened permuta- tion a bijection between two (n+1)-sets having n elements in common. A widened derangement is a widened permutation without xed points. In this paper we determine combinatorial interpretations of these func- tions in the context of the theory of species of Joyal. In particular, we prove that the species of the widened permutations is isomorphic to the derivative of the species of permutations. Looking at the generating series we obtain enumerative results, which are also obtained in a di- rect way. Finally, we prove that the sequence of widened derangement numbers turns out to coincide with the integer sequence A000255 of the On-Line Encyclopedia of Integer Sequences.