Generalized Stirling permutations and forests: Higherorder 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 threeparameter generalization of the wellknown 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 nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 DOI:
 10.48550/arXiv.1307.5624
 arXiv:
 arXiv:1307.5624
 Bibcode:
 2013arXiv1307.5624B
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics
 EPrint:
 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 ejs.sty (included). This is the journal version, with some differences with respect v2