Gonc̆arov polynomials and parking functions

Abstract

Let u be a sequence of non-decreasing positive integers. A u-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 Gonc̆arov polynomials g n ( x; a 0, a 1,…, a n−1 ) are polynomials defined by the biorthogonality relation: ε(a i)D ig n(x;a 0,a 1,…,a n−1)=n!δ in, where ε( a) is evaluation at a and D is the differentiation operator. In this paper we show that Gonc̆arov polynomials form a natural basis of polynomials for working with u-parking functions. For example, the number of u-parking functions of length n is (−1) n g n (0; u 1, u 2,…, u n ). Various properties of Gonc̆arov polynomials are discussed. In particular, Gonc̆arov polynomials satisfy a linear recursion obtained by expanding x n as a linear combination of Gonc̆arov polynomials, which leads to a decomposition of an arbitrary sequence of positive integers into two subsequences: a “maximum” u-parking function and a subsequence consisting of terms of higher values. Many counting results for parking functions can be derived from this decomposition. We give, as examples, formulas for sum enumerators, and a linear recursion and Appell relation for factorial moments of sums of u-parking functions.