For a permutation $π=π_1π_2⋯π_n𝟄 S_n$ and a positive integer $i≤ n$, we can view $π_1π_2⋯π_i$ as an element of $S_i$ by order-preserving relabeling. The $j$-set of $π$ is the set of $i$'s such that $π_1π_2⋯π_i$ is an involution in $S_i$. We prove a characterization theorem for $j$-sets, give a generating function for the number of different $j$-sets of permutations in $S_n$. We also compute the numbers of permutations in $S_n$ with a given $j$-set and prove some properties of them.