Eulerian polynomials on segmented permutations
Eulerian polynomials on segmented permutations
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发表时间：
2018-05
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通讯作者：
A. Nunge
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作者：
A. Nunge
我们定义了欧拉多项式和欧拉数的推广，考虑了研究2-物种排斥过程的分段置换的下降统计量和Hopf代数中基数的变化。给出了这些广义欧拉数所满足的一些性质。我们还定义了这些欧拉多项式的$q$-模拟，它返回了通常的欧拉多项式和有序的Bell多项式。我们还定义了一个存在于分段合成代数中的非交换类比。它给出了一个显式的母函数和一些由广义欧拉多项式所满足的恒等式，如WorPitzki型关系。
We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We give some properties satisfied by these generalized Eulerian numbers. We also define a $q$-analog of these Eulerian polynomials which gives back usual Eulerian polynomials and ordered Bell polynomials for specific values of its variables. We also define a noncommutative analog living in the algebra of segmented compositions. It gives us an explicit generating function and some identities satisfied by the generalized Eulerian polynomials such as a Worpitzky-type relation.