A formula for enumerating permutations with a fixed pinnacle set

Abstract

In 2017 Davis, Nelson, Petersen, and Tenner pioneered the study of pinnacle sets of permutations and asked whether there exists a class of operations, which applied to a permutation in Sn, can produce any other permutation with the same pinnacle set and no others. In this paper, we adapt a group action defined by Foata and Strehl to provide a way to generate all permutations with a given pinnacle set. From this we give an answer to a second question asked by Davis, Nelsen, Peterson, and Tenner, which asks for a closed non-recursive formula enumerating permutations with a given pinnacle set.