Adjacencies in Permutations

TL;DR

This paper analyzes the distribution of adjacencies in permutations and their relation to block-move operations, providing formulas for permutation counts and expected sorting moves, especially focusing on prefix transpositions.

Contribution

It introduces a method to compute the number of permutations with a given adjacency count under various block-moves and relates these counts to sorting complexity.

Findings

Cardinalities of permutation sets with specific adjacencies computed in O(n^2) time

Expected number of moves to sort permutations estimated using adjacency distributions

Results applicable to prefix and suffix transpositions due to symmetry

Abstract

A permutation on an alphabet $ \Sigma $, is a sequence where every element in $ \Sigma $ occurs precisely once. Given a permutation $ \pi $= ($\pi_{1} $, $ \pi_{2} $, $ \pi_{3} $,....., $ \pi_{n} $) over the alphabet $ \Sigma $ =$\{ $0, 1, . . . , n$-$1 $\}$ the elements in two consecutive positions in $ \pi $ e.g. $ \pi_{i} $ and $ \pi_{i+1} $ are said to form an \emph{adjacency} if $ \pi_{i+1} $ =$ \pi_{i} $+1. The concept of adjacencies is widely used in computation. The set of permutations over $ \Sigma $ forms a symmetric group, that we call P$ _{n} $. The identity permutation, I$ _{n}$ $\in$ P$_{n}$ where I$_{n}$ =(0,1,2,...,n$-$1) has exactly n$ - $1 adjacencies. Likewise, the reverse order permutation R$_{n} (\in P_{n})$=(n$-$1, n$-$2, n$-$3, n$-$4, ...,0) has no adjacencies. We denote the set of permutations in P$_{n} $ with exactly k adjacencies with P$_{n} $(k). We study variations of adjacency. % A transposition exchanges adjacent sublists; when one of the sublists is restricted to be a prefix (suffix) then one obtains a prefix (suffix) transposition. We call the operations: transpositions, prefix transpositions and suffix transpositions as block-moves. A particular type of adjacency and a particular block-move are closely related. In this article we compute the cardinalities of P$_{n}$(k) i.e. $ \forall_k \mid $P$ _{n} $ (k) $ \mid $ for each type of adjacency in $O(n^2)$ time. Given a particular adjacency and the corresponding block-move, we show that $\forall_{k} \mid P_{n}(k)\mid$ and the expected number of moves to sort a permutation in P$_{n} $ are closely related. Consequently, we propose a model to estimate the expected number of moves to sort a permutation in P$_{n} $ with a block-move. We show the results for prefix transposition. Due to symmetry, these results are also applicable to suffix transposition.