14,021 research outputs found
Reconstruction of permutations distorted by single transposition errors
The reconstruction problem for permutations on elements from their
erroneous patterns which are distorted by transpositions is presented in this
paper. It is shown that for any an unknown permutation is uniquely
reconstructible from 4 distinct permutations at transposition distance at most
one from the unknown permutation. The {\it transposition distance} between two
permutations is defined as the least number of transpositions needed to
transform one into the other. The proposed approach is based on the
investigation of structural properties of a corresponding Cayley graph. In the
case of at most two transposition errors it is shown that
erroneous patterns are required in order to reconstruct an unknown permutation.
Similar results are obtained for two particular cases when permutations are
distorted by given transpositions. These results confirm some bounds for
regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200
Growth rates of permutation classes: categorization up to the uncountability threshold
In the antecedent paper to this it was established that there is an algebraic
number such that while there are uncountably many growth
rates of permutation classes arbitrarily close to , there are only
countably many less than . Here we provide a complete characterization of
the growth rates less than . In particular, this classification
establishes that is the least accumulation point from above of growth
rates and that all growth rates less than or equal to are achieved by
finitely based classes. A significant part of this classification is achieved
via a reconstruction result for sum indecomposable permutations. We conclude by
refuting a suggestion of Klazar, showing that is an accumulation point
from above of growth rates of finitely based permutation classes.Comment: To appear in Israel J. Mat
Metric intersection problems in Cayley graphs and the Stirling recursion
In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of
elements of order 2 and let \Gamma be the Cayley graph whose vertex set is the
group G generated by H (so G is Sym(n) or Alt(n)) and whose edge set is
determined by H. We are interested in the metric structure of this graph. In
particular, for g\in G let B_{r}(g) be the metric ball in \Gamma of radius r
and centre g. We show that the intersection numbers \Phi(\Gamma; r,
g):=|\,B_{r}(e)\,\cap\,B_{r}(g)\,| are generalized Stirling functions in n and
r. The results are motivated by the study of error graphs and related
reconstruction problems.Comment: 18 page
Error Graphs and the Reconstruction of Elements in Groups
Packing and covering problems for metric spaces, and graphs in particular,
are of essential interest in combinatorics and coding theory. They are
formulated in terms of metric balls of vertices. We consider a new problem in
graph theory which is also based on the consideration of metric balls of
vertices, but which is distinct from the traditional packing and covering
problems. This problem is motivated by applications in information transmission
when redundancy of messages is not sufficient for their exact reconstruction,
and applications in computational biology when one wishes to restore an
evolutionary process. It can be defined as the reconstruction, or
identification, of an unknown vertex in a given graph from a minimal number of
vertices (erroneous or distorted patterns) in a metric ball of a given radius r
around the unknown vertex. For this problem it is required to find minimum
restrictions for such a reconstruction to be possible and also to find
efficient reconstruction algorithms under such minimal restrictions.
In this paper we define error graphs and investigate their basic properties.
A particular class of error graphs occurs when the vertices of the graph are
the elements of a group, and when the path metric is determined by a suitable
set of group elements. These are the undirected Cayley graphs. Of particular
interest is the transposition Cayley graph on the symmetric group which occurs
in connection with the analysis of transpositional mutations in molecular
biology. We obtain a complete solution of the above problems for the
transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200
Inverse monoids of partial graph automorphisms
A partial automorphism of a finite graph is an isomorphism between its vertex
induced subgraphs. The set of all partial automorphisms of a given finite graph
forms an inverse monoid under composition (of partial maps). We describe the
algebraic structure of such inverse monoids by the means of the standard tools
of inverse semigroup theory, namely Green's relations and some properties of
the natural partial order, and give a characterization of inverse monoids which
arise as inverse monoids of partial graph automorphisms. We extend our results
to digraphs and edge-colored digraphs as well
Network Inference from Co-Occurrences
The recovery of network structure from experimental data is a basic and
fundamental problem. Unfortunately, experimental data often do not directly
reveal structure due to inherent limitations such as imprecision in timing or
other observation mechanisms. We consider the problem of inferring network
structure in the form of a directed graph from co-occurrence observations. Each
observation arises from a transmission made over the network and indicates
which vertices carry the transmission without explicitly conveying their order
in the path. Without order information, there are an exponential number of
feasible graphs which agree with the observed data equally well. Yet, the basic
physical principles underlying most networks strongly suggest that all feasible
graphs are not equally likely. In particular, vertices that co-occur in many
observations are probably closely connected. Previous approaches to this
problem are based on ad hoc heuristics. We model the experimental observations
as independent realizations of a random walk on the underlying graph, subjected
to a random permutation which accounts for the lack of order information.
Treating the permutations as missing data, we derive an exact
expectation-maximization (EM) algorithm for estimating the random walk
parameters. For long transmission paths the exact E-step may be computationally
intractable, so we also describe an efficient Monte Carlo EM (MCEM) algorithm
and derive conditions which ensure convergence of the MCEM algorithm with high
probability. Simulations and experiments with Internet measurements demonstrate
the promise of this approach.Comment: Submitted to IEEE Transactions on Information Theory. An extended
version is available as University of Wisconsin Technical Report ECE-06-
Bijective enumeration of some colored permutations given by the product of two long cycles
Let be the permutation on symbols defined by $\gamma_n = (1\
2\...\ n)\betannp\gamma_n \beta^{-1}\frac{1}{n- p+1}\alpha\gamma_n\alphamn+1$, an
unexpected connection previously found by several authors by means of algebraic
methods. Moreover, our bijection allows us to refine the latter result with the
cycle type of the permutations.Comment: 22 pages. Version 1 is a short version of 12 pages, entitled "Linear
coefficients of Kerov's polynomials: bijective proof and refinement of
Zagier's result", published in DMTCS proceedings of FPSAC 2010, AN, 713-72
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