Representing Permutations with Few Moves

TL;DR

This paper studies sequences of permutations called tangles, where each element moves minimally between permutations, and proves bounds on the number of moves needed to connect any two permutations, introducing pattern-avoiding classes as a key tool.

Contribution

It establishes new bounds on the minimal moves required to connect permutations with constraints, and introduces restricted tangle classes for pattern-avoiding permutations.

Findings

Any two permutations can be connected with at most 5 moves per element.

Total moves needed can be bounded by 4n in some tangles.

Restrictions on element exchanges can reduce moves to O(log n).

Abstract

Consider a finite sequence of permutations of the elements 1,...,n, with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element i to be a maximal subsequence of at least two consecutive permutations during which its positions form an arithmetic progression of common difference +1 or -1. We prove that for any initial and final permutations, there is a tangle connecting them in which each element makes at most 5 moves, and another in which the total number of moves is at most 4n. On the other hand, there exist permutations that require at least 3 moves for some element, and at least 2n-2 moves in total. If we further require that every pair of elements exchange positions at most once, then any two permutations can be connected by a tangle with at most O(log n) moves per element, but we do not know whether this can be reduced to O(1) per element, or to O(n) in total. A key tool is the introduction of certain restricted classes of tangle that perform pattern-avoiding permutations.