Further Results on Pinnacle Sets

TL;DR

This paper advances the understanding of pinnacle sets in permutations by providing simplified proofs, recursive formulas for counting admissible orderings, and new structural insights into their properties.

Contribution

It offers a simpler combinatorial proof of a weighted sum formula and introduces recursive methods for counting permutations with given pinnacle sets.

Findings

Simplified combinatorial proof of Fang's weighted sum formula

Recursion for counting admissible orderings of pinnacle sets

New structural perspective on pinnacle sets and permutation counts

Abstract

The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis, Nelson, Petersen, and Tenner and also results proven recently by Fang, this paper aims to add to our understanding of pinnacle sets. In particular, we give a simpler and more combinatorial proof of a weighted sum formula previously proven by Fang. Additionally, we give a recursion for counting the admissible orderings of the elements of a potential pinnacle set. Finally, we give another way of viewing pinnacle sets that sheds light on their structure and also yields a recursion for counting the number of permutations with that pinnacle set.