The initial involution patterns of permutations
Summary: For a permutation \(\pi=\pi_1\pi_2\cdots\pi_n \in S_n\) and a positive integer \(i\leq n\), we can view \(\pi=\pi_1\pi_2\cdots\pi_i\) as an element of \(S_i\) by order-preserving relabeling. The \(j\)-set of \(\pi\) is the set of \(i\)'s such that \(\pi=\pi_1\pi_2\cdots\pi_i\) 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 number of permutations in \(S_n\) with a given \(j\)-set and prove some properties of them.
scientific article; zbMATH DE number 5132836