The Initial Involution Patterns of Permutations
For a permutation $\pi=\pi_1\pi_2\cdots\pi_n\in S_n$ and a positive integer $i\leq n$, we can view $\pi_1\pi_2\cdots\pi_i$ as an element of $S_i$ by order-preserving relabeling. The $j$-set of $\pi$ is the set of $i$'s such that $\pi_1\pi_2\cdots\pi_i$ 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.