Idempotents for derangement numbers Original Research Article

An analogue of the exponential generating function for derangement numbers in the symmetric group algebras is introduced. It leads to n mutually orthogonal idempotents in the group algebra of the symmetric group Sn, for all n. The corresponding representations of Sn have dimensions equal to the number of permutations with given number of fixed points. They are closely related to the higher Lie representations of Sn. As a consequence, new proofs are obtained for a number of results on derangement numbers due to Désarménien, Gessel, Reutenauer and Wachs.