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 Goncarov 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 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.

中文翻译：

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划分格和指数族中的 Gončarov 多项式

经典的 Goncarov 多项式出现在数值分析中，作为解决 Goncarov 插值问题的基础。这些多项式为停车函数的枚举理论提供了一种自然的代数工具。通过将微分算子替换为 delta 算子并使用有限算子演算理论，Lorentz、Tringali 和 Yan 引入了与 delta 算子 $\Delta$ 的一对 $(\Delta, Z)$ 相关联的广义 Goncarov 多项式序列和一个插值网格 $Z$。广义 Goncarov 多项式具有许多很好的代数性质，并且与二项式枚举和阶次统计理论有关。在本文中，我们给出了任何广义 Goncarov 多项式序列的完整组合解释。第一的，我们表明它们可以作为分区格中的权重枚举器来实现。然后，我们在指数族中给出了更具体的实现，并表明这些多项式枚举了矢量停车函数的各种丰富结构。