In this paper, we study further properties of a recently introduced generalized Eulerian number, denoted by Am;r(n; k), which reduces to the classical Eulerian number when m = 1 and r = 0. Among our results is a generalization of an earlier symmetric Eulerian number identity of Chung, Graham and Knuth. Using the row generating function for Am;r(n; k) for a fixed n, we introduce the r-Whitney-Euler-Frobenius fractions, which generalize the Euler-Frobenius fractions. Finally, we consider a further four-parameter combinatorial generalization of Am;r(n; k) and find a formula for its exponential generating function in terms of the Lambert-W function.
Bibliographical noteFunding Information:
ments which improved the article. The research of JoséL. Ramírez was partially supported by Universidad Nacional de Colombia, Project No. 46240.
© 2019 University of Belgrade.
- Combinatorial identity
- Euler-Frobenius fraction
- Eulerian number
- R-Whitney-Eulerian number
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics