Let \(S_{n}\) denote the symmetric group of order n. Say that two subsets \(x, y\subseteq S_{n}\) are 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 Tripathi, 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 \(3\le k\le n!-3\) , where \(\Lambda _n(k)=\frac{1}{n!^2}\left( {\begin{array}{c}n!\\ k\end{array}}\right) \) . Furthermore, we prove that \( \frac{1}{\Lambda _n(n!/2)}T\!\left( n,\left[ \sqrt{\tfrac{n!}{4}}x+\tfrac{n!}{2}\right] \right) ~\xrightarrow {n\rightarrow \infty }~ \exp \!\left( -\tfrac{x^2}{2}\right) \,, \) uniformly over \(\mathbb {R}\) .
