Some aspects of (r, k)-parking functions

摘要

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) = center dot center dot center dot = b(n) satisfies b(i) = 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 (sic)(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 (Sigma(n = 0) F(n)((r,1))t(n))(k) for any k is an element of Z. When k 0, this power series is just Sigma(n = 0) F-n((r,k)) t(n); 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 F-n((r,1)) to define a multiplicative basis for the ring Lambda of symmetric functions. We investigate some of the properties of this basis. (C) 2018 Elsevier Inc. All rights reserved.