Flattened Stirling Permutations

TL;DR

This paper introduces flattened Stirling permutations, establishes a bijection with type B set partitions, and explores their combinatorial properties, including enumeration and maximum run counts.

Contribution

It defines flattened Stirling permutations, links them to type B set partitions, and provides enumeration results and bounds on their structural features.

Findings

Bijection between flattened Stirling permutations and type B set partitions.

Enumeration of flattened Stirling permutations with few runs.

Maximum number of runs in flattened Stirling permutations.

Abstract

Recall that a Stirling permutation is a permutation on the multiset $\{1,1,2,2,\ldots,n,n\}$ such that any numbers appearing between repeated values of $i$ must be greater than $i$. We call a Stirling permutation ``flattened'' if the leading terms of maximal chains of ascents (called runs) are in weakly increasing order. Our main result establishes a bijection between flattened Stirling permutations and type $B$ set partitions of $\{0,\pm1,\pm2,\ldots,\pm (n-1)\}$, which are known to be enumerated by the Dowling numbers, and we give an independent proof of this fact. We also determine the maximal number of runs for any flattened Stirling permutation, and we enumerate flattened Stirling permutations with a small number of runs or with two runs of equal length. We conclude with some conjectures and generalizations worthy of future investigation.