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- W4297805392 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." @default.
- W4297805392 created "2022-10-01" @default.
- W4297805392 creator A5083778034 @default.
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- W4297805392 date "2016-01-18" @default.
- W4297805392 modified "2023-09-28" @default.
- W4297805392 title "Adjacencies in Permutations" @default.
- W4297805392 doi "https://doi.org/10.48550/arxiv.1601.04469" @default.
- W4297805392 hasPublicationYear "2016" @default.
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