Adjacencies in Permutations

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.Comment: 20 pages. 5 table

Sorting permutations with a transposition tree

The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $\pi$ and $\pi$ is in $S_n$. $T$ is the operation and the edge set $E_T$ denotes the corresponding generator set. The goal is to sort a given permutation $\pi$ with $T$. The number of generators of $E_T$ that suffices to sort any $\pi \in S_n$ constitutes an upper bound. It is an upper bound, on the diameter of the corresponding Cayley graph $\Gamma$ i.e. $diam(\Gamma)$. A precise upper bound equals $diam(\Gamma)$. Such bounds are known only for a few trees. Jerrum showed that computing $diam(\Gamma)$ is intractable in general if the number of generators is two or more whereas $T$ has $n-1$ generators. For several operations computing a tight upper bound is of theoretical interest. Such bounds have applications in evolutionary biology to compute the evolutionary relatedness of species and parallel/distributed computing for latency estimation. The earliest algorithm computed an upper bound $f(\Gamma)$ in a $\Omega(n!)$ time by examining all $\pi$ in $S_n$. Subsequently, polynomial time algorithms were designed to compute upper bounds or their estimates. We design an upper bound $\delta^*$ whose cumulative value for all trees of a given size $n$ is shown to be the tightest for $n \leq 15$. We show that $\delta^*$ is tightest known upper bound for full binary trees. Keywords: Transposition trees, Cayley graphs, permutations, sorting, upper bound, diameter, greedy algorithms.Comment: 13 pages. 4 figures, 5 table