On the enumeration of parking functions by leading numbers

Let x = (x ₁,..., x _n) be a sequence of positive integers. An x-parking function is a sequence (a ₁,..., a _n) of positive integers whose non-decreasing rearrangement b ₁ ≤ ⋯ ≤b _n satisfies b _i ≤ x ₁ + ⋯ + x _i. 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.