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

Jesús Salas,Eduardo J.S. Villaseñor,J. Fernando Barbero G.

doi:10.37236/4814

Jesús Salas, Eduardo J.S. Villaseñor + Show 1 more

Open Access

https://doi.org/10.37236/4814

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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.

Highlights

Stirling permutations of order n are permutations of the multiset {12, 22, . . . , n2} such that, for each 1 r n, the elements appearing between two occurrences of r are at least r [16]
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
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 provide a combinatorial interpretation

Summary

Introduction

Stirling permutations of order n are permutations of the multiset {12, 22, . . . , n2} such that, for each 1 r n, the elements appearing between two occurrences of r are at least r [16]. Second order Eulerian numbers are closely related to Ward numbers Wn,k [30], [22, entry A134991]. They form an inverse pair in the sense of Riordan [27] (see[28], [22, entry A008517]): Wn,k = Bn,k =. Stirling permutations and Eulerian numbers have been generalized to multisets of the form {1ν, 2ν, . Some particular numbers of this class have been considered in other contexts [4, 12, 5, 20] but, to our knowledge, most of them have not appeared before in the literature. These relations provide a simple combinatorial interpretation for the generalized Ward numbers in the present context

Generalized Stirling permutations

Increasing trees and forests

Combinatorial interpretation of the generalized Ward numbers