# The pinnacle set of a permutation

## TL;DR

This paper introduces the concept of pinnacle sets, which record the values of peaks in permutations, and explores their properties, characterizations, and enumerative aspects, highlighting differences from traditional peak sets.

## Contribution

It defines and characterizes admissible pinnacle sets of permutations and investigates their enumerative properties, providing new insights into permutation peak structures.

## Key findings

- Characterization of admissible pinnacle sets
- Enumeration formulas for pinnacle sets
- Differences between pinnacle and peak set distributions

## Abstract

The peak set of a permutation records the indices of its peaks. These sets have been studied in a variety of contexts, including recent work by Billey, Burdzy, and Sagan, which enumerated permutations with prescribed peak sets. In this article, we look at a natural analogue of the peak set of a permutation, instead recording the values of the peaks. We define the "pinnacle set" of a permutation w to be the set {w(i) : i is a peak of w}. Although peak sets and pinnacle sets mark the same phenomenon for a given permutation, the behaviors of these sets differ in notable ways as distributions over the symmetric group. In the work below, we characterize admissible pinnacle sets and study various enumerative questions related to these objects.