Growth Rates Of Permutations With Given Descent Or Peak Set

TL;DR

This paper studies the asymptotic growth rates of permutations with fixed descent or peak sets, establishing the possible ranges and density of these rates across all such sets.

Contribution

It characterizes the exact range of growth rates for descent sets and shows the density of peak set growth rates within a specific interval.

Findings

Growth rates for descent sets cover the entire interval [0, 2/π].

Growth rates for periodic peak sets are dense in [0, 1/∛3].

Algorithms are provided to construct sets with desired growth rates.

Abstract

Given a set $I \subseteq \mathbb{N}$, consider the sequences $\{d_n(I)\},\{p_n(I)\}$ where for any $n$, $d_n(I)$ and $p_n(I)$ respectively count the number of permutations in the symmetric group $\mathfrak{S}_n$ whose descent set (respectively peak set) is $I \cap [n-1]$. We investigate the growth rates $\text{gr} \ d_n(I) = \lim_{n \to \infty} \left(d_n(I)/n!\right)^{1/n}$ and $\text{gr} \ p_n(I) = \lim_{n \to \infty} \left(p_n(I)/n!\right)^{1/n}$ over all $I \subseteq \mathbb{N}$. Our main contributions are two-fold. Firstly, we prove that the numbers $\text{gr} \ d_n(I)$ over all $I \subseteq \mathbb{N}$ are exactly the interval $\left[0,2/\pi\right]$. To do so, we construct an algorithm that explicitly builds $I$ for any desired limit $L$ in the interval. Secondly, we prove that the numbers $\text{gr} \ p_n(I)$ for periodic sets $I \subseteq \mathbb{N}$ form a dense set in $\left[0,1/\sqrt[3]{3}\right]$. We do this by explicitly finding, for any prescribed $L$ in the interval, a set $I$ whose corresponding growth rate is arbitrarily close to $L$.