The operators [formula omitted] on permutations, 132-avoiding permutations and inversions

Abstract

We introduce the operators Fi on permutation π=π1⋯πk−11πk+1⋯πn of {1,2,…,n}, where 1≤i≤k−1, i.e., define Fi(π)=π1′π2′⋯πn′ as πj′=πj−1 for 1≤j≤i, and πi+1′πi+2′⋯πn′ has the same relative order as πi+1πi+2⋯πn. The operators Fi have many properties concerning the 132-pattern and inversions. Furthermore, we find that the operators Fi can be characterized by a series of swaps of two entries. Two applications of the operators are given. As a first application, we obtain some new objects in 132-avoiding permutations and in Dyck paths that are enumerated by the entries in Catalan’s triangle. As another application, we give an algorithm to generate the set of permutations of length n+1 with k inversions from the set of permutations of length n with k inversions when n≥k+1.