Some aspects of (r,k)-parking functions

Abstract

An (r,k)-parking function of length n may be defined as a sequence (a1,…,an) of positive integers whose increasing rearrangement b1≤⋯≤bn satisfies bi≤k+(i−1)r. The case r=k=1 corresponds to ordinary parking functions. We develop numerous properties of (r,k)-parking functions. In particular, if Fn(r,k) denotes the Frobenius characteristic of the action of the symmetric group Sn on the set of all (r,k)-parking functions of length n, then we find a combinatorial interpretation of the coefficients of the power series (∑n≥0Fn(r,1)tn)k for any k∈Z. When k>0, this power series is just ∑n≥0Fn(r,k)tn; when k<0, we obtain a dual to (r,k)-parking functions. We also give a q-analogue of this result. For fixed r, we can use the symmetric functions Fn(r,1) to define a multiplicative basis for the ring Λ of symmetric functions. We investigate some of the properties of this basis.