5,905 research outputs found
Combinatorial Enumeration of Graphs
In this chapter, I will talk about some of the enumerative combinatorics problems that have interested researchers during the last decades. For some of those enumeration problems, it is possible to obtain closed mathematical expressions, and for some other it is possible to obtain an estimation by the use of asymptotic methods. Some of the methods used in both cases will be covered in this chapter as well as some application of graph enumeration in different fields. An overview about the enumeration of trees will be given as an example of combinatorial problem solved in a closed mathematical form. Similarly, the problem of enumeration of regular graphs will be discussed as an example of combinatorial enumeration for which it is hard to obtain a closed mathematical form solution and apply the asymptotic estimation method used frequently in analytic combinatorics for this end. An example of application of the enumerative combinatorics for obtaining a result of applicability criteria of selection nodes in a virus spreading control problem will be given as well
Bounded affine permutations I. Pattern avoidance and enumeration
We introduce a new boundedness condition for affine permutations, motivated
by the fruitful concept of periodic boundary conditions in statistical physics.
We study pattern avoidance in bounded affine permutations. In particular, we
show that if is one of the finite increasing oscillations, then every
-avoiding affine permutation satisfies the boundedness condition. We also
explore the enumeration of pattern-avoiding affine permutations that can be
decomposed into blocks, using analytic methods to relate their exact and
asymptotic enumeration to that of the underlying ordinary permutations.
Finally, we perform exact and asymptotic enumeration of the set of all bounded
affine permutations of size . A companion paper will focus on avoidance of
monotone decreasing patterns in bounded affine permutations.Comment: 35 page
Where two fractals meet: the scaling of a self-avoiding walk on a percolation cluster
The scaling properties of self-avoiding walks on a d-dimensional diluted
lattice at the percolation threshold are analyzed by a field-theoretical
renormalization group approach. To this end we reconsider the model of Y. Meir
and A. B. Harris (Phys. Rev. Lett. 63:2819 (1989)) and argue that via
renormalization its multifractal properties are directly accessible. While the
former first order perturbation did not agree with the results of other
methods, we find that the asymptotic behavior of a self-avoiding walk on the
percolation cluster is governed by the exponent nu_p=1/2 + epsilon/42 +
110epsilon^2/21^3, epsilon=6-d. This analytic result gives an accurate numeric
description of the available MC and exact enumeration data in a wide range of
dimensions 2<=d<=6.Comment: 4 pages, 2 figure
The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics
Prudent walks are special self-avoiding walks that never take a step towards
an already occupied site, and \emph{-sided prudent walks} (with )
are, in essence, only allowed to grow along directions. Prudent polygons
are prudent walks that return to a point adjacent to their starting point.
Prudent walks and polygons have been previously enumerated by length and
perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration
of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find
that the generating function is expressed in terms of a -hypergeometric
function, with an accumulation of poles towards the dominant singularity. This
expression reveals an unusual asymptotic structure of the number of polygons of
area , where the critical exponent is the transcendental number
and and the amplitude involves tiny oscillations. Based on numerical data, we
also expect similar phenomena to occur for 4-sided polygons. The asymptotic
methodology involves an original combination of Mellin transform techniques and
singularity analysis, which is of potential interest in a number of other
asymptotic enumeration problems.Comment: 27 pages, 6 figure
On the enumeration of closures and environments with an application to random generation
Environments and closures are two of the main ingredients of evaluation in
lambda-calculus. A closure is a pair consisting of a lambda-term and an
environment, whereas an environment is a list of lambda-terms assigned to free
variables. In this paper we investigate some dynamic aspects of evaluation in
lambda-calculus considering the quantitative, combinatorial properties of
environments and closures. Focusing on two classes of environments and
closures, namely the so-called plain and closed ones, we consider the problem
of their asymptotic counting and effective random generation. We provide an
asymptotic approximation of the number of both plain environments and closures
of size . Using the associated generating functions, we construct effective
samplers for both classes of combinatorial structures. Finally, we discuss the
related problem of asymptotic counting and random generation of closed
environemnts and closures
Asymptotic formulas for stacks and unimodal sequences
We study enumeration functions for unimodal sequences of positive integers,
where the size of a sequence is the sum of its terms. We survey known results
for a number of natural variants of unimodal sequences, including Auluck's
generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions.
These results describe combinatorial properties, generating functions, and
asymptotic formulas for the enumeration functions. We also prove several new
asymptotic results that fill in the notable missing cases from the literature,
including an open problem in statistical mechanics due to Temperley.
Furthermore, we explain the combinatorial and asymptotic relationship between
partitions, Andrews' Frobenius symbols, and stacks with summits.Comment: 19 pages, 4 figure
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