Pinnacle sets revisited

TL;DR

This paper advances the combinatorics of pinnacles in permutations by providing a simple recursion for counting permutations with a given pinnacle set, proposing a conjectural formula, and analyzing the structure of related permutation orbits.

Contribution

It introduces an efficient recursion for computing permutation counts with specified pinnacle sets and characterizes the structure of permutation orbits under the modified Foata-Strehl action.

Findings

Developed a simple recursion for $p_n(S)$

Proposed a conjectural closed formula for $q_n(S)$

Characterized and counted minimal and maximal elements in permutation orbits

Abstract

In 2017, Davis, Nelson, Petersen, and Tenner [Discrete Math. 341 (2018),3249--3270] initiated the combinatorics of pinnacles in permutations. We provide a simple and efficient recursion to compute $p_n(S)$, the number of permutations of $S_n$ with pinnacle set $S$, and a conjectural closed formula for the related numbers $q_n(S)$. We determine the lexicographically minimal elements of the orbits of the modified Foata-Strehl action, prove that these elements form a lower ideal of the left weak order and characterize and count the maximal elements of this ideal.