Generalized Stirling Permutations and Forests: Higher-Order Eulerian and Ward Numbers

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.We are indebted to Alan Sokal for his participation in the early stages of this work, his encouragement, and useful suggestions later on. We also thank Jesper Jacobsen, Anna de Mier, Neil Sloane, and Mike Spivey for correspondence, and David Callan for pointing out some interesting references to us. Last but not least, we thank Bojan Mohar for valuable criticisms and suggestions