论文标题
Dumont Ansatz用于欧拉的多项式，峰值多项式和导数多项式
The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials
论文作者
论文摘要
我们观察到，通过Ma-Ma-Yeh提出的语法转变的想法，可以将杜蒙的三种无上下文语法带入共同点。然后，我们开发了一个统一的观点，以研究与双变量欧拉尔多项式相关的几个组合对象。我们将这种方法称为杜蒙特·安萨兹（Dumont Ansatz）。 作为应用，我们本着符号方法的精神，春季数量的关系，欧拉数字，三种峰值多项式，彼得森的身份以及两种衍生性多项式，由Knuth-Buckholtz和Carlitz-Scoville引入的两种衍生性多项式，提供语法处理。我们在左峰多项式上获得卷积公式，导致Gessel公式。在此框架中，由于Josuat-Vergès，我们被导致对衍生多项式的组合解释。
We observe that three context-free grammars of Dumont can be brought to a common ground, via the idea of transformations of grammars, proposed by Ma-Ma-Yeh. Then we develop a unified perspective to investigate several combinatorial objects in connection with the bivariate Eulerian polynomials. We call this approach the Dumont ansatz. As applications, we provide grammatical treatments, in the spirit of the symbolic method, of relations on the Springer numbers, the Euler numbers, the three kinds of peak polynomials, an identity of Petersen, and the two kinds of derivative polynomials, introduced by Knuth-Buckholtz and Carlitz-Scoville, and later by Hoffman in a broader context. We obtain a convolution formula on the left peak polynomials, leading to the Gessel formula. In this framework, we are led to the combinatorial interpretations of the derivative polynomials due to Josuat-Vergès.
