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 $\mathbb{ R}$.
Categories: math.CO math.PR
