Year: 2025
Type: article
Abstract: This paper studies higher-order Eulerian numbers based on Stirling permutations and utilizing Eulerian triangles. It primarily focuses on the chain of higher-order Eulerian numbers, higher-order Eulerian polynomials, and higher-order Eulerian fractions, especially their computation. Many results for Eulerian numbers and second-order Eulerian numbers are generalized to higher-order Eulerian numbers. More specifically, we present recurrence relations of high-order Eulerian numbers, row-generating functions, and row sums of higher-order Eulerian triangles. Furthermore, we investigate the higher-order Eulerian fraction and its alternative form. Some properties of higher-order Eulerian fractions are expressed using differentiation and integration. We derive the inversion relations between second-order Eulerian numbers and Stirling numbers of the second and first kinds. Finally, we provide exact expressions and a computational method for higher-order Eulerian numbers.
Source: Axioms
Author: Tian-Xiao He
Institution: Illinois Wesleyan University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,896
OpenAlex ID: W7117293580
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