Let x=(x1,...,xn) be a sequence of positive integers. An x-parking function is a sequence (a1,...,an) of positive integers whose non-decreasing rearrangement b1≤⋯≤bn satisfies bi≤x1+⋯+xi. In this paper we give a combinatorial approach to the enumeration of (a,b,...,b) -parking functions by their leading terms, which covers the special cases x=(1,...,1), (a,1,...,1), and (b,...,b). The approach relies on bijections between the x-parking functions and labeled rooted forests. To serve this purpose, we present a simple method for establishing the required bijections. Some bijective results between certain sets of x-parking functions of distinct leading terms are also given. © 2005 Elsevier Inc. All rights reserved.
Eu, S. P., Fu, T. S., & Lai, C. J. (2005). On the enumeration of parking functions by leading terms. Advances in Applied Mathematics, 35(4), 392–406. https://doi.org/10.1016/j.aam.2005.03.005