On the enumeration of parking functions by leading terms

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.