Enumeration of a dual set of Stirling permutations by their alternating runs

TL;DR

This paper investigates a dual set of Stirling permutations, focusing on counting them based on their alternating runs, and explores their generating functions, recurrence relations, and combinatorial properties.

Contribution

It introduces a new dual set of Stirling permutations and analyzes their properties through generating functions and recurrence relations.

Findings

Derived recurrence relations for the dual Stirling permutations

Established grammatical interpretations of the generating functions

Presented convolution formulas related to the enumeration

Abstract

In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.