On the enumeration of parking functions by leading terms

Abstract

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.