Number of orbits of $k$-subsets of permutations

TL;DR

This paper investigates the asymptotic number of subsets of the symmetric group under an equivalence relation, revealing a Gaussian limit distribution for the count of subsets of size around half of the total permutations.

Contribution

It provides asymptotic formulas for the number of permutation subsets up to equivalence and establishes a Gaussian limit law for their distribution.

Findings

Asymptotic formula for T(n,k) as rac{1}{n!^2}inom{n!}{k}

Gaussian limit distribution for normalized subset counts

Exact counts for small subset sizes (0, 1, 2)

Abstract

Let $S_n$ denote the symmetric group of order $n$. Say that two subsets $x, y\subseteq S_n$ are \emph{equivalent} if there exist permutations $g_1, g_2\in S_n$ such that $g_1xg_2=y$, where multiplication is understood elementwise. Recently, [Tripathi, 2024] and [Kushwaha and Triathi, 2025] asked for the asymptotics of $T(n,k)$, the number of subsets of $S_n$ of size $k$ up to this equivalence. It is easy to see that $T(n,0)=T(n, 1)=1$ and $T(n, 2)=p(n)-1$, where $p(n)$ is the number of integer partitions of $n$. In this work, we show that $T(n,k) = \Lambda_n(k)(1+o_n(1))$ for $3\leq k\leq n!-3$, where $\Lambda_n(k)=\frac{1}{n!^2}\binom{n!}{k}$. Furthermore, we prove that $$\frac{1}{\Lambda_n(n!/2)}T\!\left(n,\left[\sqrt{\tfrac{n!}{4}}x+\tfrac{n!}{2}\right]\right) ~\xrightarrow{n\to\infty}~ \exp\!\left(-\tfrac{x^2}{2}\right),$$ uniformly over $\mathbb{R}$.