Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin