Gončarov polynomials and parking functions

• Authors: Joseph P. S. Kung;Catherine Yan
• Venue: Journal of Combinatorial Theory Series A
• Year: 2003

Let u be a sequence of non-decreasing positive integers. A u-parking function of length n is a sequence (x1, x2, ..., xn) whose order statistics (the sequence (x(1), x(2), ..., x(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x(i) ≤ ui. The Goncarov polynomials gn(x; a0, a1, ..., an-1) are polynomials defined by the biorthogonality relation: ε(ai)Dign(x; a0, a1, ..., an-1 = n!δin, where ε(a) is evaluation at a and D is the differentiation operator. In this paper we show that Goncarov 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 gn(O; u1, u2, ..., un). Various properties of Goncarov polynomials are discussed. In particular, Goncarov polynomials satisfy a linear recursion obtained by expanding xn as a linear combination of Goncarov 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.