Download or read book Patterns in Permutations and Words written by Sergey Kitaev and published by Springer Science & Business Media. This book was released on 2011-08-30 with total page 511 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been considerable interest recently in the subject of patterns in permutations and words, a new branch of combinatorics with its roots in the works of Rotem, Rogers, and Knuth in the 1970s. Consideration of the patterns in question has been extremely interesting from the combinatorial point of view, and it has proved to be a useful language in a variety of seemingly unrelated problems, including the theory of Kazhdan—Lusztig polynomials, singularities of Schubert varieties, interval orders, Chebyshev polynomials, models in statistical mechanics, and various sorting algorithms, including sorting stacks and sortable permutations. The author collects the main results in the field in this up-to-date, comprehensive reference volume. He highlights significant achievements in the area, and points to research directions and open problems. The book will be of interest to researchers and graduate students in theoretical computer science and mathematics, in particular those working in algebraic combinatorics and combinatorics on words. It will also be of interest to specialists in other branches of mathematics, theoretical physics, and computational biology. The author collects the main results in the field in this up-to-date, comprehensive reference volume. He highlights significant achievements in the area, and points to research directions and open problems. The book will be of interest to researchers and graduate students in theoretical computer science and mathematics, in particular those working in algebraic combinatorics and combinatorics on words. It will also be of interest to specialists in other branches of mathematics, theoretical physics, and computational biology.
Download or read book Words and Permutations Avoiding and Satisfying Contiguous Patterns written by T.A. Tshifhumulo and published by . This book was released on 2003 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Permutation Patterns written by Steve Linton and published by Cambridge University Press. This book was released on 2010-06-03 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: A mixture of survey and research articles by leading experts that will be of interest to specialists in permutation patterns and other researchers in combinatorics and related fields. In addition, the volume provides plenty of material accessible to advanced undergraduates and is a suitable reference for projects and dissertations.
Download or read book Counting Patterns in Permutations and Words written by Jeffrey Edward Liese and published by . This book was released on 2008 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of permutations and permutation statistics dates back hundreds of years to the time of Euler and before. In this thesis, we examine several generalizations of classical permutation statistics, most often generalizing the descent statistic, des(sigma). Chapter 1 is dedicated to providing some history and background to the work presented in later chapters. Chapter 2 reviews permutations, notations and the study of several classic permutation statistics. It is interesting to note that many surprising identities and connections to other areas of combinatorics arise as we refine the descent statistic. In Chapter 3, we consider a more refined pattern matching condition where we take into account conditions involving the equivalence classes of the elements of a descent mod k for some integer k>̲ 2. In general, when one includes parity conditions or conditions involving equivalence mod k, then the problem of counting the number of pattern matchings becomes more complicated. We then proceed to provide q-analogues to these findings and present them in Chapter 4. In Chapter 5, we prove some results on patterns in words. In particular we show that the generating functions for words embedding specific patterns are rational functions. In fact we also develop a method to obtain these generating functions using a finite state automaton. Thus, we can compare generating functions for words embedding different patterns. Sometimes these generating functions are the same, so many bijective questions arise from this study. We will then review some work of Jeff Remmel and Anthony Mendes. In particular, they were able to find generating functions which count occurrences of consecutive sequences in a permutation or a word which matches a given pattern by exploiting the combinatorics associated with symmetric functions. They were able to take the generating function for the number of permutations which do not contain a certain pattern and give generating functions refining permutations by both the total number of pattern matches and the number of non-overlapping pattern matches. However, as a corollary, the generating function that they produced involved a term counting the number of permutations that have consecutive overlapping patterns at certain positions. We begin to enumerate these for permutations in S4 and S5 in Chapter 6. Lastly, we look at yet another generalization of the descent statistic where we require the descent to be equal to a fixed value, k. Our results in this area are presented in Chapter 7.
Download or read book Words and Graphs written by Sergey Kitaev and published by Springer. This book was released on 2015-11-18 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first comprehensive introduction to the theory of word-representable graphs, a generalization of several classical classes of graphs, and a new topic in discrete mathematics. After extensive introductory chapters that explain the context and consolidate the state of the art in this field, including a chapter on hereditary classes of graphs, the authors suggest a variety of problems and directions for further research, and they discuss interrelations of words and graphs in the literature by means other than word-representability. The book is self-contained, and is suitable for both reference and learning, with many chapters containing exercises and solutions to seleced problems. It will be valuable for researchers and graduate and advanced undergraduate students in discrete mathematics and theoretical computer science, in particular those engaged with graph theory and combinatorics, and also for specialists in algebra.
Download or read book Generalized Patterns in Words and Permutations written by Sergey Kitaev and published by . This book was released on 2003 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Combinatorics of Compositions and Words written by Silvia Heubach and published by CRC Press. This book was released on 2009-07-20 with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt: A One-Stop Source of Known Results, a Bibliography of Papers on the Subject, and Novel Research Directions Focusing on a very active area of research in the last decade, Combinatorics of Compositions and Words provides an introduction to the methods used in the combinatorics of pattern avoidance and pattern enumeration in compositions and words. It
Download or read book Avoiding and Enforcing Repetitive Structures in Words written by Mike Müller and published by BoD – Books on Demand. This book was released on 2014-12-15 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: Avoiding and enforcing repetitions in words are central topics in the area of combinatorics on words, with first results going back to the beginning of the 20th century. The results presented in this thesis extend and enrich the existing theory concerning the presence and absence of repetitive structures in words. In the first part the question whether such structures necessarily appear in infinite words over a finite alphabet is investigated. In particular, avoidability questions of patterns whose repetitive structure is disguised by the application of a permutation are studied. The second part deals with equations on words that enforce a certain repetitive structure involving involutions in their solution set. A generalisation of the classical equations u^l = v^mw^n that were studied by Lyndon and Schützenberger is analysed. The last part considers the influence of the shuffle operation on square-free words and related avoidability questions.
Download or read book Surveys in Combinatorics 2013 written by Simon R. Blackburn and published by Cambridge University Press. This book was released on 2013 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: Surveys of recent important developments in combinatorics covering a wide range of areas in the field.
Download or read book Combination Optimization of Subsequence Patterns in Words written by Matthew R. Just and published by . This book was released on 2016 with total page 89 pages. Available in PDF, EPUB and Kindle. Book excerpt: Author's abstract: Packing patterns in words concerns finding a word with the maximum number of a prescribed pattern. The majority of the work done thus far is on packing patterns into permutations. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently, the packing density for all but two (up to equivalence) permutation patterns up to length 4 can be obtained. In this thesis we consider the analogous question for colored patterns and permutations. By introducing the concept of colored blocks we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples, we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities.
Download or read book Discrete Mathematics written by Oscar Levin and published by Createspace Independent Publishing Platform. This book was released on 2018-07-30 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: Note: This is a custom edition of Levin's full Discrete Mathematics text, arranged specifically for use in a discrete math course for future elementary and middle school teachers. (It is NOT a new and updated edition of the main text.)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.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 Algebraic Combinatorics on Words written by M. Lothaire and published by Cambridge University Press. This book was released on 2002-04-18 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: Comprehensive 2002 introduction to combinatorics on words for mathematicians and theoretical computer scientists.
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 Enumeration Schemes for Pattern avoiding Words and Permutations written by and published by . This book was released on 2008 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let p=p_1 ... p_n and q=q_1 ... q_m be permutations. We say that p contains q as a pattern if there are indices 1
Download or read book Combinatorial Pattern Matching written by Bin Ma and published by Springer Science & Business Media. This book was released on 2007-06-22 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume features select refereed proceedings from the 18th Annual Symposium on Combinatorial Pattern Matching. Collectively, the papers provide great insights into the most recent advances in combinatorial pattern matching. They are organized into topical sections covering algorithmic techniques, approximate pattern matching, data compression, computational biology, pattern analysis, suffix arrays and trees, and algorithmic techniques.
Download or read book Developments in Language Theory written by Igor Potapov and published by Springer. This book was released on 2015-07-17 with total page 459 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 19th International Conference on Developments in Language Theory, DLT 2015, held in Liverpool, UK. The 31 papers presented together with 5 invited talks were carefully reviewed and selected from 54 submissions. Its scope is very general and includes, among others, the following topics and areas: combinatorial and algebraic properties of words and languages, grammars, acceptors and transducers for strings, trees, graphs, arrays, algebraic theories for automata and languages, codes, efficient text algorithms, symbolic dynamics, decision problems, relationships to complexity theory and logic, picture description and analysis, polyominoes and bidimensional patterns, cryptography, concurrency, cellular automata, bio-inspired computing, and quantum computing.
Download or read book Block Patterns in Permutations and Words and Generalized Clusters written by Ran Pan and published by . This book was released on 2016 with total page 189 pages. Available in PDF, EPUB and Kindle. Book excerpt: Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words without consecutive repeats, colored permutations without consecutive repeated colors, Carlitz integer compositions, Young tableaux, non-backtracking random walks, ordered set partitions, cycle structures in permutations and so on. We develop an extension of the cluster method which we call the generalized cluster method to study the distribution of certain consecutive patterns in such restricted combinatorial objects. The generalized cluster method enables us to express the generating function for distribution of a pattern in such restricted combinatorial objects in terms of so-called generalized cluster polynomials. Compared to the original problem, computing generalized cluster polynomials is usually more tractable. We also generalize a multi-variate version of both cluster method and generalized cluster method which is used to study joint distribution of multiple patterns. We use combinatorial objects mentioned above as concrete examples to demonstrate our method.
