# Gon\v{c}arov Polynomials in Partition Lattices and Exponential Families

## TL;DR

This paper provides a combinatorial interpretation of generalized Goncarov polynomials, connecting them to partition lattices and exponential families, and demonstrating their enumeration of enriched vector parking functions.

## Contribution

It introduces a complete combinatorial interpretation of generalized Goncarov polynomials, linking them to partition lattices and exponential families, and explores their enumeration of enriched parking functions.

## Key findings

- Goncarov polynomials can be realized as weight enumerators in partition lattices.
- They can be concretely realized in exponential families.
- They enumerate various enriched structures of vector parking functions.

## Abstract

Classical Gon\v{c}arov polynomials arose in numerical analysis as a basis for the solutions of the Gon\v{c}arov 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 Gon\v{c}arov polynomials associated to a pair $(\Delta, Z)$ of a delta operator $\Delta$ and an interpolation grid $Z$. Generalized Gon\v{c}arov polynomials share many nice algebraic properties and have a connection with the theories of binomial enumeration and order statistics. In this paper we give a complete combinatorial interpretation for any sequence of generalized Gon\v{c}arov 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.