Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers
Abstract
We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a Riordan inverse pair, for which we also provide a combinatorial interpretation.
Cite
@article{arxiv.1307.5624,
title = {Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers},
author = {J. Fernando Barbero G. and Jesús Salas and Eduardo J. S. Villaseñor},
journal= {arXiv preprint arXiv:1307.5624},
year = {2021}
}
Comments
19 pages (LaTeX2e). v2: Many changes with respect to v1: old section 3 corresponds to new sections 4 and 5, and the other sections in the new version contain new combinatorial interpretations of these numbers. v3: uses e-js.sty (included). This is the journal version, with some differences with respect v2