The skew, relative, and classical derangements

In his popular combinatories text, Brualdi elucidates the principle of inclusion and exclusion with the classical and the relative derangements. Eventually, the two kinds of derangements are linked up via an algebraic relationship from the parallel use of the principle of inclusion and exclusion. We introduce the notion of skew derangements and relate them to relative derangements and the classical derangements by a purely combinatorial correspondence. Moreover, with the aid of our bijection we easily generalize the relative derangements, obtaining a binomial-type formula for the number of such generalized relative derangements on n elements in terms of the classical derangement number.