On ^-derangement Numbers

We derive a (/-analogue of the classical formula for the number of derangements of an n element set. Our derivation is entirely analogous to the classical derivation, but relies on a descent set preserving bijection between the set of permutations with a given derangement part and the set of shuffles of two permutations. A classical application of binomial inversion (more generally the principle of inclusion-exclusion) is the derivation of the formula for the number of derangements (permutations with no fixed points) of an zz element set: " t uk k=0 This is obtained by counting permutations according to their number of fixed points and then inverting the resulting equation. In this note we shall derive a formula of I. Gessel [G] for ^-counting derangements by the major index statistic in a way entirely analogous to the classical q = 1 case. That is, we shall #-count permutations with k fixed points and then use Gauss inversion ( ̂ -binomial inversion or more generally Möbius inversion on the lattice of subspaces of a vector space) to derive the following formula for ^-derangement numbers: k=0 L J' A key step in our derivation and an interesting result in its own right is a descent-preserving bijection between the set of permutations with a given derangement part and the set of shuffles of two permutations. This bijection enables us to use a formula of A. Garsia and I. Gessel for ^-counting shuffles. Gessel [G] obtained the formula for ^-derangement numbers as a corollary of an Eulerian generating function formula for counting permutations by descents, major index, and cycle structure, which is proved via a correspondence Received by the editors February 6, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 05A30; Secondary 05A19, 05A15, 05A05. Research partially supported by NSF grant DMS:8503700. ©1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page