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 Goncharov polynomials
gn(x; a0,a1,...,an-1) are polynomials
defined by the biorthogonality relation:
Goncharov polynomials also satisfy a linear recursion obtained by expanding xn as a linear combination of Goncharov polynomials. The combinatorial structure underlying this recursion is 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. From this combinatorial decomposition, we derive linear recursions for sum enumerators, expected sums of u-parking functions, and higher moments of sums of u-parking functions. These recursions yield explicit formulas for these quantities in terms of Goncharov polynomials.
This is joint work with Catherine Yan.