Download or read book Lattice Path Counting and Applications written by Gopal Mohanty and published by Academic Press. This book was released on 2014-07-10 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Lattice Path Counting and Applications focuses on the principles, methodologies, and approaches involved in lattice path counting and applications, including vector representation, random walks, and rank order statistics. The book first underscores the simple and general boundaries of path counting. Topics include types of diagonal steps and a correspondence, paths within general boundaries, higher dimensional paths, vector representation, compositions, and domination, recurrence and generating function method, and reflection principle. The text then examines invariance and fluctuation and random walk and rank order statistics. Discussions focus on random walks, rank order statistics, Chung-Feller theorems, and Sparre Andersen's equivalence. The manuscript takes a look at convolution identities and inverse relations and discrete distributions, queues, trees, and search codes, as well as discrete distributions and a correlated random walk, trees and search codes, convolution identities, and orthogonal relations and inversion formulas. The text is a valuable reference for mathematicians and researchers interested in in lattice path counting and applications.

Download or read book Lattice Path Combinatorics and Applications written by George E. Andrews and published by Springer. This book was released on 2019-03-02 with total page 443 pages. Available in PDF, EPUB and Kindle. Book excerpt: The most recent methods in various branches of lattice path and enumerative combinatorics along with relevant applications are nicely grouped together and represented in this research contributed volume. Contributions to this edited volume will be mainly research articles however it will also include several captivating, expository articles (along with pictures) on the life and mathematical work of leading researchers in lattice path combinatorics and beyond. There will be four or five expository articles in memory of Shreeram Shankar Abhyankar and Philippe Flajolet and honoring George Andrews and Lajos Takács. There may be another brief article in memory of Professors Jagdish Narayan Srivastava and Joti Lal Jain. New research results include the kernel method developed by Flajolet and others for counting different classes of lattice paths continues to produce new results in counting lattice paths. The recent investigation of Fishburn numbers has led to interesting counting interpretations and a family of fascinating congruences. Formulas for new methods to obtain the number of Fq-rational points of Schubert varieties in Grassmannians continues to have research interest and will be presented here. Topics to be included are far reaching and will include lattice path enumeration, tilings, bijections between paths and other combinatoric structures, non-intersecting lattice paths, varieties, Young tableaux, partitions, enumerative combinatorics, discrete distributions, applications to queueing theory and other continuous time models, graph theory and applications. Many leading mathematicians who spoke at the conference from which this volume derives, are expected to send contributions including. This volume also presents the stimulating ideas of some exciting newcomers to the Lattice Path Combinatorics Conference series; “The 8th Conference on Lattice Path Combinatorics and Applications” provided opportunities for new collaborations; some of the products of these collaborations will also appear in this book. This book will have interest for researchers in lattice path combinatorics and enumerative combinatorics. This will include subsets of researchers in mathematics, statistics, operations research and computer science. The applications of the material covered in this edited volume extends beyond the primary audience to scholars interested queuing theory, graph theory, tiling, partitions, distributions, etc. An attractive bonus within our book is the collection of special articles describing the top recent researchers in this area of study and documenting the interesting history of who, when and how these beautiful combinatorial results were originally discovered.

Download or read book Counting Lattice Paths Using Fourier Methods written by Shaun Ault and published by Springer Nature. This book was released on 2019-08-30 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference. Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.

Download or read book Analytic Combinatorics written by Philippe Flajolet and published by Cambridge University Press. This book was released on 2009-01-15 with total page 825 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.

Download or read book Lattice Path Combinatorics and Special Counting Sequences written by Chunwei Song and published by CRC Press. This book was released on 2024-09-17 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book endeavors to deepen our understanding of lattice path combinatorics, explore key types of special sequences, elucidate their interconnections, and concurrently champion the author's interpretation of the “combinatorial spirit”. The author intends to give an up-to-date introduction to the theory of lattice path combinatorics, its relation to those special counting sequences important in modern combinatorial studies, such as the Catalan, Schröder, Motzkin, Delannoy numbers, and their generalized versions. Brief discussions of applications of lattice path combinatorics to symmetric functions and connections to the theory of tableaux are also included. Meanwhile, the author also presents an interpretation of the "combinatorial spirit" (i.e., "counting without counting", bijective proofs, and understanding combinatorics from combinatorial structures internally, and more), hoping to shape the development of contemporary combinatorics. Lattice Path Combinatorics and Special Counting Sequences: From an Enumerative Perspective will appeal to graduate students and advanced undergraduates studying combinatorics, discrete mathematics, or computer science.

Download or read book Discrete Mathematics written by Oscar Levin and published by Createspace Independent Publishing Platform. This book was released on 2016-08-16 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions.

Download or read book Combinatorics The Art of Counting written by Bruce E. Sagan and published by American Mathematical Soc.. This book was released on 2020-10-16 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a gentle introduction to the enumerative part of combinatorics suitable for study at the advanced undergraduate or beginning graduate level. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature which has never before appeared in print, such as the use of quotient posets to study the Möbius function and characteristic polynomial of a partially ordered set, or the connection between quasisymmetric functions and pattern avoidance. The book assumes minimal background, and a first course in abstract algebra should suffice. The exposition is very reader friendly: keeping a moderate pace, using lots of examples, emphasizing recurring themes, and frankly expressing the delight the author takes in mathematics in general and combinatorics in particular.

Download or read book Bijective Combinatorics written by Nicholas Loehr and published by CRC Press. This book was released on 2011-02-10 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bijective proofs are some of the most elegant and powerful techniques in all of mathematics. Suitable for readers without prior background in algebra or combinatorics, Bijective Combinatorics presents a general introduction to enumerative and algebraic combinatorics that emphasizes bijective methods.The text systematically develops the mathematical

Download or read book The Major Counting of Nonintersecting Lattice Paths and Generating Functions for Tableaux written by Christian Krattenthaler and published by American Mathematical Soc.. This book was released on 1995 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: A theory of counting nonintersecting lattice paths by the major index and its generalizations is developed. We obtain determinantal expressions for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given final points, where the starting points lie on a line parallel to [italic]x + [italic]y = 0. In some cases these determinants can be evaluated to result in simple products. As applications we compute the generating function for tableaux with [italic]p odd rows, with at most [italic]c columns, and with parts between 1 and [italic]n. Moreover, we compute the generating function for the same kind of tableaux which in addition have only odd parts. We thus also obtain a closed form for the generating function for symmetric plane partitions with at most [italic]n rows, with parts between 1 and [italic]c, and with [italic]p odd entries on the main diagonal. In each case the result is a simple product. By summing with respect to [italic]p we provide new proofs of the Bender-Knuth and MacMahon (ex-)conjectures, which were first proved by Andrews, Gordon, and Macdonald. The link between nonintersecting lattice paths and tableaux is given by variations of the Knuth correspondence.

Download or read book Analytic Combinatorics in Several Variables written by Robin Pemantle and published by Cambridge University Press. This book was released on 2013-05-31 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed at graduate students and researchers in enumerative combinatorics, this book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective.

Download or read book An Invitation to Analytic Combinatorics written by Stephen Melczer and published by Springer Nature. This book was released on 2020-12-22 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book uses new mathematical tools to examine broad computability and complexity questions in enumerative combinatorics, with applications to other areas of mathematics, theoretical computer science, and physics. A focus on effective algorithms leads to the development of computer algebra software of use to researchers in these domains. After a survey of current results and open problems on decidability in enumerative combinatorics, the text shows how the cutting edge of this research is the new domain of Analytic Combinatorics in Several Variables (ACSV). The remaining chapters of the text alternate between a pedagogical development of the theory, applications (including the resolution by this author of conjectures in lattice path enumeration which resisted several other approaches), and the development of algorithms. The final chapters in the text show, through examples and general theory, how results from stratified Morse theory can help refine some of these computability questions. Complementing the written presentation are over 50 worksheets for the SageMath and Maple computer algebra systems working through examples in the text.

Download or read book Walk Through Combinatorics A An Introduction To Enumeration And Graph Theory Third Edition written by Miklos Bona and published by World Scientific Publishing Company. This book was released on 2011-05-09 with total page 567 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.Just as with the first two editions, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible to the talented and hardworking undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings, Eulerian and Hamiltonian cycles, and planar graphs.The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs (new to this edition), enumeration under group action (new to this edition), generating functions of labeled and unlabeled structures and algorithms and complexity.As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.The Solution Manual is available upon request for all instructors who adopt this book as a course text. Please send your request to [email protected].

Download or read book A Course in Enumeration written by Martin Aigner and published by Springer Science & Business Media. This book was released on 2007-06-28 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt: Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from basic notions of combinatorial enumeration to a variety of topics, ranging from algebra to statistical physics. The book is organized in three parts: Basics, Methods, and Topics. The aim is to introduce readers to a fascinating field, and to offer a sophisticated source of information for professional mathematicians desiring to learn more. There are 666 exercises, and every chapter ends with a highlight section, discussing in detail a particularly beautiful or famous result.

Download or read book Algebraic Geometry for Scientists and Engineers written by Shreeram Shankar Abhyankar and published by American Mathematical Soc.. This book was released on 1990 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, this book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities.

Download or read book Handbook of Enumerative Combinatorics written by Miklos Bona and published by CRC Press. This book was released on 2015-03-24 with total page 1073 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting the state of the art, the Handbook of Enumerative Combinatorics brings together the work of today's most prominent researchers. The contributors survey the methods of combinatorial enumeration along with the most frequent applications of these methods.This important new work is edited by Miklos Bona of the University of Florida where he

Download or read book The q t Catalan Numbers and the Space of Diagonal Harmonics written by James Haglund and published by American Mathematical Soc.. This book was released on 2008 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work contains detailed descriptions of developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. These developments have led in turn to some surprising discoveries in the combinatorics of Macdonald polynomials.

Download or read book Computing the Continuous Discretely written by Matthias Beck and published by Springer. This book was released on 2015-11-14 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device. The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more. With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume? Reviews of the first edition: “You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.” — MAA Reviews “The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate rial, exercises, open problems and an extensive bibliography.” — Zentralblatt MATH “This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.” — Mathematical Reviews “Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.” — CHOICE