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Classical Gon\v{c}arov polynomials arose in numerical analysis as a basis forthe solutions of the Gon\v{c}arov interpolation problem. These polynomialsprovide a natural algebraic tool in the enumerative theory of parkingfunctions. By replacing the differentiation operator with a delta operator andusing the theory of finite operator calculus, Lorentz, Tringali and Yanintroduced the sequence of generalized Gon\v{c}arov polynomials associated to apair $(\Delta, Z)$ of a delta operator $\Delta$ and an interpolation grid $Z$.Generalized Gon\v{c}arov polynomials share many nice algebraic properties andhave a connection with the theories of binomial enumeration and orderstatistics. In this paper we give a complete combinatorial interpretation forany sequence of generalized Gon\v{c}arov polynomials. First, we show that theycan be realized as weight enumerators in partition lattices. Then, we give amore concrete realization in exponential families and show that thesepolynomials enumerate various enriched structures of vector parking functions.

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