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Adjacencies in Permutations
A permutation on an alphabet , is a sequence where every element in
occurs precisely once. Given a permutation = (, , ,....., ) over the alphabet =0, 1, . . . , n1 the elements in two consecutive positions in
e.g. and are said to form an \emph{adjacency} if =+1. The concept of adjacencies is widely used in
computation. The set of permutations over forms a symmetric group,
that we call P. The identity permutation, I P where
I =(0,1,2,...,n1) has exactly n1 adjacencies. Likewise, the
reverse order permutation R=(n1, n2, n3, n4,
...,0) has no adjacencies. We denote the set of permutations in P with
exactly k adjacencies with P(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(k) i.e. P (k) for
each type of adjacency in time. Given a particular adjacency and the
corresponding block-move, we show that and the
expected number of moves to sort a permutation in P are closely related.
Consequently, we propose a model to estimate the expected number of moves to
sort a permutation in P 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