On Path diagrams and Stirling permutations

TL;DR

This paper extends the concept of local types from permutations to $k$-Stirling permutations, establishing bijections with increasing trees and deriving continued fraction representations of their generating functions.

Contribution

It generalizes the notion of local types to $k$-Stirling permutations and provides new branched continued fraction representations via path diagram bijections.

Findings

Bijection between local types and node types of $(k+1)$-ary increasing trees

Branched continued fraction representations of generating functions

Multiple continued fraction representations for Stirling permutations

Abstract

A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutations). We establish a bijection of these local types to node types of $(k+1)$-ary increasing trees. We present a branched continued fraction representation of the generating function of these local types through a bijection with path diagrams induced by \L ukasiewicz paths, generalizing the results from permutations to arbitrary $k$-Stirling permutations. We further show that the generating function of ordinary Stirling permutation has at least three branched continued fraction representations, using correspondences between non-standard increasing trees, $k$-Stirling permutations and path diagrams.