Number of orbits of $k$-subsets of permutations

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 $T(n,k) = Λ_n(k)(1+o_n(1))$ for $3\leq k\leq n!-3$, where $Λ_n(k)=\frac{1}{n!^2}\binom{n!}{k}$. Furthermore, we prove that $$\frac{1}{Λ_n(n!/2)}T\!\left(n,\left[\sqrt{\tfrac{n!}{4}}x+\tfrac{n!}{2}\right]\right) ~\xrightarrow{n\to\infty}~ \exp\!\left(-\tfrac{x^2}{2}\right),$$ uniformly over $\mathbb{R}$.