Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers

TL;DR

This paper introduces a new family of generalized Stirling permutations linked to ordered trees and forests, extending Eulerian numbers and connecting them to Ward numbers through Riordan arrays.

Contribution

It defines generalized Stirling permutations, derives their generating functions, and establishes a combinatorial link to Ward numbers via Riordan inverse pairs.

Findings

Number of permutations with fixed ascents given by three-parameter Eulerian generalization

Closed formulas for simple cases and row polynomials

Establishment of a Riordan inverse pair connecting 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.