Extending the parking space

The action of the symmetric group S n on the set Park n of parking functions of size n has received a great deal of attention in algebraic combinatorics. We prove that the action of S n on Park n extends to an action of S n + 1 . More precisely, we construct a graded S n + 1 -module V n such that the restriction of V n to S n is isomorphic to Park n . We describe the S n -Frobenius characters of the module V n in all degrees and describe the S n + 1 -Frobenius characters of V n in extreme degrees. We give a bivariate generalization V n ( ℓ , m ) of our module V n whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.