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

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 $(\mathrm{\Delta },Z)$ of a delta operator $\mathrm{\Delta }$ 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.