Some aspects of (r,k)-parking functions

Richard Stanley, Yinghui Wang

Published 2016-04-27Version 1

An (r,k)-parking function} of length n may be defined as a sequence (a_1,...,a_n) of positive integers whose increasing rearrangement b_1\leq ... \leq b_n satisfies b_i\leq 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 F_n^{(r,k)} denotes the Frobenius characteristic of the action of the symmetric group S_n 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 (\sum_{n\geq 0}F_n^{(r,1)}t^n)^k for any integer k. For instance, when k>0 this power series is just \sum_{n\geq 0} F_n^{(r,k)} t^n. We also give a q-analogue of this result. For fixed r, we can use the symmetric functions F_n^{(r,1)} to define a multiplicative basis for the ring of symmetric functions. We investigate some of the properties of this basis.