640 research outputs found
The Sortability of Graphs and Matrices under Context Directed Swaps
The study of sorting permutations by block interchanges has recently been
stimulated by a phenomenon observed in the genome maintenance of certain
ciliate species. The result was the identification of a block interchange
operation that applies only under certain constraints. Interestingly, this
constrained block interchange operation can be generalized naturally to simple
graphs and to an operation on square matrices. This more general context
provides numerous techniques applicable to the original context. In this paper
we consider the more general context, and obtain an enumeration, in closed
form, of all simple graphs on n vertices that are ``sortable" by the graph
analogue of the constrained version of block interchanges. We also obtain
asymptotic results on the proportion of graphs on n vertices that are so
sortable
Classifying Permutations under Context-Directed Swaps and the \textbf{cds} game
A special sorting operation called Context Directed Swap, and denoted
\textbf{cds}, performs certain types of block interchanges on permutations.
When a permutation is sortable by \textbf{cds}, then \textbf{cds} sorts it
using the fewest possible block interchanges of any kind. This work introduces
a classification of permutations based on their number of \textbf{cds}-eligible
contexts. In prior work an object called the strategic pile of a permutation
was discovered and shown to provide an efficient measure of the
non-\textbf{cds}-sortability of a permutation. Focusing on the classification
of permutations with maximal strategic pile, a complete characterization is
given when the number of \textbf{cds}-eligible contexts is close to maximal as
well as when the number of eligible contexts is minimal. A group action that
preserves the number of \textbf{cds}-eligible contexts of a permutation
provides, via the orbit-stabilizer theorem, enumerative results regarding the
number of permutations with maximal strategic pile and a given number of
\textbf{cds}-eligible contexts. Prior work introduced a natural two-person game
on permutations that are not \textbf{cds}-sortable. The decision problem of
which player has a winning strategy in a particular instance of the game
appears to be of high computational complexity. Extending prior results, this
work presents new conditions for player ONE to have a winning strategy in this
combinatorial game.Comment: 22 page
A simple framework on sorting permutations
In this paper we present a simple framework to study various distance
problems of permutations, including the transposition and block-interchange
distance of permutations as well as the reversal distance of signed
permutations. These problems are very important in the study of the evolution
of genomes. We give a general formulation for lower bounds of the transposition
and block-interchange distance from which the existing lower bounds obtained by
Bafna and Pevzner, and Christie can be easily derived. As to the reversal
distance of signed permutations, we translate it into a block-interchange
distance problem of permutations so that we obtain a new lower bound.
Furthermore, studying distance problems via our framework motivates several
interesting combinatorial problems related to product of permutations, some of
which are studied in this paper as well.Comment: 13 pages. This is the second part from division of the paper:
arXiv:1411.5552v2 [math.CO], into two parts. The first part is:
arXiv:1502.07674 [math.CO]. The original paper arXiv:1411.5552v2 [math.CO]
will be removed soon. Comments are welcome. [v2]:Theorem 3 has been
generalized to arbitrary permutation
Context Directed Reversals and the Ciliate Decryptome
Prior studies of the efficiency of the block interchange (swap) and the
reversal sorting operations on (signed) permutations identified specialized
versions of the these operations. These specialized operations are here called
context directed reversal, abbreviated cdr, and context directed swap,
abbreviated cds. Prior works have also characterized which (signed)
permutations are sortable by cdr or by cds.
It is now known that when a permutation is cds sortable in n steps, then any
application of n consecutive applicable cds operations will sort it. Examples
show that this is not the case for cdr. This phenomenon is the focus of this
paper. It is proven that if a signed permutation is cdr sortable, then any cdr
fixed point of it is cds sortable (the cds Rescue Theorem). The cds Rescue
Theorem is discussed in the context of a mathematical model for ciliate
micronuclear decryption.
It is also proven that though for a given permutation the number of cdr
operations leading to different cdr fixed points may be different from each
other, the parity of these two numbers is the same (the cdr Parity Theorem).
This result provides a solution to two previously formulated decision problems
regarding certain combinatorial games.Comment: 18 pages, 13 figure
Plane permutations and applications to a result of Zagier-Stanley and distances of permutations
In this paper, we introduce plane permutations, i.e. pairs
where is an -cycle and is an arbitrary
permutation, represented as a two-row array. Accordingly a plane permutation
gives rise to three distinct permutations: the permutation induced by the upper
horizontal (), the vertical ) and the diagonal () of
the array. The latter can also be viewed as the three permutations of a
hypermap. In particular, a map corresponds to a plane permutation, in which the
diagonal is a fixed point-free involution. We study the transposition action on
plane permutations obtained by permuting their diagonal-blocks. We establish
basic properties of plane permutations and study transpositions and exceedances
and derive various enumerative results. In particular, we prove a recurrence
for the number of plane permutations having a fixed diagonal and cycles in
the vertical, generalizing Chapuy's recursion for maps filtered by the genus.
As applications of this framework, we present a combinatorial proof of a result
of Zagier and Stanley, on the number of -cycles , for which the
product has exactly cycles. Furthermore, we
integrate studies on the transposition and block-interchange distance of
permutations as well as the reversal distance of signed permutations. Plane
permutations allow us to generalize and recover various lower bounds for
transposition and block-interchange distances and to connect reversals with
block-interchanges.Comment: To appear in SIAM J. Discrete Math. Considering the scope of the
journal, the content in arXiv:1502.07971 "A simple framework on sorting
permutations" was include
Using Ciliate Operations to construct Chromosome Phylogenies
We develop an algorithm based on three basic DNA editing operations suggested
by a model for ciliate micronuclear decryption, to transform a given
permutation into another. The number of ciliate operations performed by our
algorithm during such a transformation is taken to be the distance between two
such permutations. Applying well-known clustering methods to such distance
functions enables one to determine phylogenies among the items to which the
distance functions apply. As an application of these ideas we explore the
relationships among the chromosomes of eight fruitfly (drosophila) species,
using the well-known UPGMA algorithm on the distance function provided by our
algorithm.Comment: 31 pages, 14 figures. Preliminary repor
On a lower bound for sorting signed permutations by reversals
Computing the reversal distances of signed permutations is an important topic
in Bioinformatics. Recently, a new lower bound for the reversal distance was
obtained via the plane permutation framework. This lower bound appears
different from the existing lower bound obtained by Bafna and Pevzner through
breakpoint graphs. In this paper, we prove that the two lower bounds are equal.
Moreover, we confirm a related conjecture on skew-symmetric plane permutations,
which can be restated as follows: let
and let
be any long cycle on the set . Then,
and are always in the same cycle of the product .
Furthermore, we show the new lower bound via plane permutations can be
interpreted as the topological genera of orientable surfaces associated to
signed permutations.Comment: slightly update
Sorting a Permutation by block moves
We prove a lower and an upper bound on the number of block moves necessary to
sort a permutation. We put our results in contrast with existing results on
sorting by block transpositions, and raise some open questions.Comment: 7 page
Permutation sorting and a game on graphs
We introduce a game on graphs. By a theorem of Zermelo, each instance of the
game on a finite graph is determined. While the general decision problem on
which player has a winning strategy in a given instance of the game is
unsolved, we solve the decision problem for a specific class of finite graphs.
This result is then applied to a permutation sorting game to prove the
optimality of a proportional bound under which TWO has a winning strategy.Comment: 14 page
Quantifying CDS Sortability of Permutations by Strategic Pile Size
The special purpose sorting operation, context directed swap (CDS), is an
example of the block interchange sorting operation studied in prior work on
permutation sorting. CDS has been postulated to model certain molecular sorting
events that occur in the genome maintenance program of some species of
ciliates. We investigate the mathematical structure of permutations not
sortable by the CDS sorting operation. In particular, we present substantial
progress towards quantifying permutations with a given strategic pile size,
which can be understood as a measure of CDS non-sortability. Our main results
include formulas for the number of permutations in with maximum
size strategic pile. More generally, we derive a formula for the number of
permutations in with strategic pile size , in addition to an
algorithm for computing certain coefficients of this formula, which we call
merge numbers.Comment: 19 page
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