Analytic combinatorics

This chapter introduces analytic combinatorics, a modern approach to the study of combinatorial structures of the sort that we encounter frequently in the analysis of algorithms. The approach is predicated on the idea that combinatorial structures are typically defined by simple formal rules that are the key to learning their properties.
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The purpose of this chapter is largely to serve as an accessible introduction or a refresher of basic notions regarding analytic functions. We start by recalling the elementary theory of functions and their singularities in a style tuned to the needs of analytic combinatorics. Cauchy’s integral formula expresses coefficients of analytic functions as contour integrals. Suitable uses of Cauchy’s integral formula then make it possible to estimate such coefficients by suitably selecting an appropriate contour of integration.
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Chapter 3: Generating Functions introduces a central concept in the averagecase analysis of algorithms: generating functions  a necessary and natural link between the algorithms that are our objects of study and analytic methods that are necessary to discover their properties.
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Chapter 9: Words and maps covers global properties of words (Nletter strings from an Mletter alphabet), which are wellstudied in classical combinatorics (because they model sequences of independent Bernoulli trials) and in classical applied algorithmics (because they model input sequences for hashing algorithms). The chapter also covers random maps (Nletter words from an Nletter alphabet) and discusses relationships with trees and permutations.
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In this chapter, we examine some of the most important classes of labelled objects, including surjections, set partitions, permutations, as well as labelled graphs, trees, and mappings from a finite set into itself. Certain aspects of words can also be treated by this theory, a fact which has important consequences not only in combinatorics itself but also in probability and statistics.
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The primary goal of this chapter is to provide combinatorial illustrations of the power of complex analytic methods, and specifically of the rational–meromorphic framework developed in the previous chapter. At the same time, we shift gears and envisage counting problems at a new level of generality.
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A saddlepoint of a surface is a point reminiscent of the inner part of a saddle or of a geographical pass between two mountains. If the surface represents the modulus of an analytic function, saddlepoints are simply determined as the zeros of the derivative of the function. This chapter presents the following content: Modulus surfaces, saddle point bounds, saddle point asymptotics, applications.
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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Airy Phenomena and Analytic Combinatorics of Connected Graphs...
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Chapter 1: Analysis of Algorithms considers the general motivations for algorithmic analysis and relationships among various approaches to studying performance characteristics of algorithms. This chapter includes contents: History and motivation, a scientific approach, example: Quicksort, resources.
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Chapter 2: Recurrence Relations concentrates on fundamental mathematical properties of various types of recurrence relations which arise frequently when analyzing an algorithm through a direct mapping from a recursive representation of a program to a recursive representation of a function describing its properties.
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This chapter examines methods of deriving approximate solutions to problems or of approximating exact solutions, which allow us to develop concise and precise estimates of quantities of interest when analyzing algorithms.
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Chapter 6: Trees investigates properties of many different types of trees, fundamental structures that arise implicitly and explicitly in many practical algorithms. Our goal is to provide access to results from an extensive literature on the combinatorial analysis of trees, while at the same time providing the groundwork for a host of algorithmic applications.
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Chapter 7: Permutations surveys combinatorial properties of permutations (orderings of the numbers 1 through N) and shows how they relate in a natural way to fundamental and widelyused sorting algorithms.
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Chapter 8: String and Tries studies basic combinatorial properties of strings, sequences of characters or letters drawn from a fixed alphabet, and introduces algorithms that process strings ranging from fundamental methods at the heart of the theory of computation to practical textprocessing methods with a host of important applications.
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This chapter and the next are devoted to enumeration, where the problem is to determine the number of combinatorial configurations described by finite rules, and do so for all possible sizes. This chapter presents the following content: Symbolic method, trees and strings, powersets and multisets, compositions and partitions, substitution.
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Many scientific endeavours demand precise quantitative information on probabilistic properties of parameters of combinatorial objects. This chapter introduce to combinatorial parameters and MGFs. This chapter presents the following content: Basics, moment calculations, OBGF examples, labelled classes.
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In this chapter, we present a general approach to the analysis of coefficients of generating functions that is not restricted to polar singularities and extends to a large class of functions that have moderate growth or decay at their dominant singularities. This chapter presents the following content: Prelude, standard function scale, singularity analysis, schemas and transfer theorems.
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Singularity analysis paves the way to the analysis of a large quantity of generating functions, as provided by the symbolic method expounded in Chapters I–III. In this chapter we illustrate this situation with numerous examples related to languages, permutations, trees, and graphs of various sorts. As in chapter 5, most analyses are organized into broad classes called schemas.
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Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we ﬁnd that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples. ...
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