Expected Sums of General Parking Functions

Abstract

A (u ₁, u ₂, . . . )-parking function of length n is a sequence (x ₁, x ₂, . . . , x _n) whose order statistics (the sequence (x ₍₁₎, x ₍₂₎, . . . , x _(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x _(i)≤ u _(i). The Gončarov polynomials g _n (x; a ₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals ε(a _i ) D ⁱ , where ε(a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gončarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i =a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u _i+1 - u _i . Finally, we give analogues of our results for real-valued parking functions.