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

Abstract

Listen

Translate article

We introduce the operators F i on permutation π = π 1 ⋯ π k − 1 1 π k + 1 ⋯ π n of { 1 , 2 , … , n } , where 1 ≤ i ≤ k − 1 , i.e., define F i ( π ) = π 1 ′ π 2 ′ ⋯ π n ′ as π j ′ = π j − 1 for 1 ≤ j ≤ i , and π i + 1 ′ π i + 2 ′ ⋯ π n ′ has the same relative order as π i + 1 π i + 2 ⋯ π n . The operators F i have many properties concerning the 132-pattern and inversions. Furthermore, we find that the operators F i 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 .