Exact formulas for moments of sums of classical parking functions

Abstract

For given positive integers a and b, an [ a, b]-parking function of length n is a sequence ( x 1, x 2,…, x n ) of positive integers whose order statistics x ( i) (the sequence obtained by rearranging the original sequence in non-decreasing order) satisfy the inequalities x ( i) ⩽ a+( i−1) b for all i. Using elementary methods, we derive explicit formulas for higher moments of sums and reversed sums of parking functions of length n. These formulas are finite double sums with reasonably simple terms. They are obtained by solving a recursion based on a combinatorial decomposition which decomposes a sequence of positive integers into a “maximum” parking function and a subsequence all of whose terms have “high” values. Moments of reversed sums of ordinary parking functions (the case when a= b=1) have a surprising connection with the enumeration of “sparsely-edged” graphs. From this connection, we obtain exact formulas and derive, using routine methods, asymptotic formulas (due originally to E.M. Wright) for the number of connected labeled graphs on N vertices and N+ k edges. Our method yields a new formula for the Wright constants in terms of a sequence which satisfies a linear recursion.