The $m$th-order Eulerian Numbers
Combinatorics
2023-12-29 v1
Abstract
We define the th-order Eulerian numbers with a combinatorial interpretation. The recurrence relation of the th-order Eulerian numbers, the row generating function and the row sums of the th-order Eulerian triangle are presented. We also define the th-order Eulerian fraction and its alternative form. Some properties of the th-order Eulerian fractions are represented by using differentiation and integration. An inversion relationship between second-order Eulerian numbers and Stirling numbers of the second kind is given. Finally, we give the exact expression of the values of the th-order Eulerian numbers.
Keywords
Cite
@article{arxiv.2312.17153,
title = {The $m$th-order Eulerian Numbers},
author = {Tian-Xiao He},
journal= {arXiv preprint arXiv:2312.17153},
year = {2023}
}