Efficient recurrence for the enumeration of permutations with fixed pinnacle set

Wenjie Fang

doi:10.46298/dmtcs.8321

Abstract

Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute efficiently the number $|\mathfrak{S}_n(P)|$ of permutations of size $n$ with a given pinnacle set $P$, with arithmetic complexity $O(k^4 + k\log n)$ for $P$ of size $k$. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum $q_n(P)$ of $|\mathfrak{S}_n(P)|$ proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple form, and a conjectural form is given recently by Flaque, Novelli and Thibon (2021+). We settle the problem by providing and proving an alternative form of $q_n(P)$, which has a strong combinatorial flavor. We also study admissible orderings of a given pinnacle set, first considered by Rusu (2020) and characterized by Rusu and Tenner (2021), and we give an efficient algorithm for their counting.

Highlights

Given a permutation π ∈ Sn in one-line notation π1π2 · · · πn, we consider its local maxima, i.e., elements πi for 2 ≤ i ≤ n − 1 such that πi−1 < πi > πi+1
The pinnacle set of a permutation π, denoted by Pin(π), is the set of pinnacles of π
For P ⊆ [n], we denote by Sn(P ) the set of permutations with pinnacle set P

Summary

Introduction

Given a permutation π ∈ Sn in one-line notation π1π2 · · · πn, we consider its local maxima, i.e., elements πi for 2 ≤ i ≤ n − 1 such that πi−1 < πi > πi+1. Using the same recurrence in Theorem 4.1, through a computer algebra system, we obtain general symbolic expressions of |Sn(P )| for P of arbitrary fixed size, extending formulas given in Davis et al (2018) for |P | = 1, 2.

Construction of permutations with a fixed pinnacle set

Motzkin path compression

Counting pinnacle orders