Goncarov Polynomials, Partition Lattices and Parking Sequences

Abstract

Classical Goncarov polynomials arose in numerical analysis as a basis for the solutions of the Goncarov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introduced the sequence of generalized Goncarov polynomials associated to a pair $(\Delta, Z)$ of a delta operator $\Delta$ and an interpolation grid $Z$. Generalized \gon polynomials share many nice algebraic properties and have a connection with the theories of binomial enumeration and order statistics. Parking functions are combinatorial objects which were introduced in 1966 by Konheim and Weiss. They have been well-studied in the literature due to their numerous connections and have several generalizations and extensions. Ehrenborg and Happ recently introduced a generalization of parking functions called parking sequences in which the $n$ cars have different sizes, and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. Consequently, this dissertation is divided into two major parts. In the first part, we give a complete combinatorial interpretation for any sequence of generalized Goncarov polynomials. First, we show that they can be realized as weight enumerators in partition lattices. Then, we give a more concrete realization in exponential families and show that these polynomials enumerate various enriched structures of vector parking functions. In the second part, we study increasing parking sequences and their representation via a special class of lattice paths. We also study two notions of invariance in parking sequences and prove some interesting results for a number of cases where the sequence of car sizes have special properties.