DOCSLIB.ORG

Manipulating Combinatorial Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Manipulating Combinatorial Structures DOCUMENT RESUME ED 472 095 SE 066 382 AUTHOR Labelle, Gilbert TITLE Manipulating Combinatorial Structures. PUB DATE 2000-00-00 NOTE 18p.; In: Canadian Mathematics Education Study Group = Groupe Canadien d'Etude en Didactique des Mathematiques. Proceedings of the Annual Meeting (24th, Montreal, Quebec, Canada, May 26-30, 2000); see SE 066 379. PUB TYPE Guides Non-Classroom (055) -- Speeches/Meeting Papers (150) EDRS PRICE EDRS Price MF01/PC01 Plus Postage. DESCRIPTORS Curriculum Development; Higher Education; Interdisciplinary Approach; Mathematical Applications; *Mathematical Formulas; Mathematical Models; *Mathematics Instruction; Modern Mathematics; Teaching Methods IDENTIFIERS Combinatorics ABSTRACT This set of transparencies shows how the manipulation of combinatorial structures in the context of modern combinatorics can easily lead to interesting teaching and learning activities at every level of education from elementary school to university. The transparencies describe: (1) the importance and relations of combinatorics to science and social activities;(2) how analytic/algebraic combinatorics is similar to analytic/algebraic geometry;(3) basic combinatorial structures with terminology;(4) species of structures;(5) power series associated with any species of structures;(6) similarity of operations between power series and the corresponding species of structures;(7) examples illustrating where the structures are manipulated; and (8) general teaching/learning activities. (KHR) Reproductions supplied by EDRS are the best that can be made from the original document. PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS O BEEN GRANTED BY Manipulating Combinatorial Structures Gilbert Labelle TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) Universite du Quebec a Montreal (UQAM) 1 A combinatorial structure is a finite construction made on a finite set of elements. In real-life U.S. DEPARTMENT OF EDUCATION situations, combinatorial structures often arise as "skeletons" or "schematic descriptions" Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION of concrete objects. For example, on a road map, the elements can be cities and the finite CENTER (ERIC) This document has been reproduced as construction can be the various roads joining these cities. Similarly, a diamond can be con- received from the person or organization originating it. sidered as a combinatorial structure: the plane facets of the diamond are connected together Minor changes have been made to according to certain rules. improve reproduction quality. Combinatorics can be defined as the mathematical analysis, classification and enu- meration of combinatorial structures. The main purpose of my presentation is to show how Points of view or opinions stated in this document do not necessarily represent the manipulation of combinatorial structures, in the context of modern combinatorics, can official OERI position cr policy. easily lead to interesting teaching/learning activities at every level of education: from el- ementary school to university. The following pages of these proceedings contain a grayscale version of my (color) transparencies (two transparencies on each page). I decided to publish directly my trans- parencies (instead of a standard typed text) for two main reasons: my talk contained a great amount of figures, which had to be reproduced anyway, and the short sentences in the transparencies are easy to read and put emphasis on the main points I wanted to stress. The first two transparencies are preliminary ones and schematically describe: a) the importance and relations of combinatorics with science/ social activities, b) how analytic/ algebraic combinatorics is similar to analytic/ algebraic geometry. The next 7 transparencies (numbered 1 to 7) contain drawings showing basic combinatorial structures together with some terminology. Collecting together similar combinatorial structures give rise to the con- cept of species of structures (transparencies 8 to 12). A power series is then associated to any species of structures enabling one to count its structures (transparencies 13 to 15). Each op- eration on power series (sum, product, substitution, derivation) is reflected by similar opera- tions on the corresponding species of structures (transparencies 16 to 18). The power of this correspondence is illustrated on explicit examples (transparencies 18 to 26) where the struc- tures are manipulated (and counted) using various combinatorial operations. Transparen- cies 27 to 30 suggest some general teaching/ learning activities (from elementary to advanced levels) that may arise from these ideas. I hope that the readers will agree that manipulating combinatorial structures constitutes a good activity to develop the mathematical mind. References The main references for the theory of combinatorial species are: Joyal, A. (1981). Une theorie combinatoire des series formelles, Advances in Mathematics, 42, 1-82. Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial Species and Tree-like Structures. Encyclopedia of Math. and its Applications, Vol. 67, Cambridge University Press. BEST COPYAVAILABLE 2 3 The dynamicPRELIMINARIES connections between GeometricalANALYTIC / ALGEBRAIC MathematicalGEOMETRY combinatorialwill greatlymathematics increase / discrete duringand the comingscienceactivities decades I social : figures -4 2xformulas - ey +7 = 0 Computercompression(analysis of algorithms. of data science structures, organization etc) and 0 x2+y2= 9 (classification.Chemistry symmetries of molecules, y=e2 isomers.quantum(classification,Physics etc)theory. symmetries Feynman integrals. of atoms, etc) ANALYTIC / ALGEBRAIC COMBINATORICS Combinatorics classification(analysisBiology of DNA, of species, genetic code.etc) Combinatorialstructures 4-----> Mathematicalformulas graphs,(analysis,Mathematics groups, calculus, etc) algebra, geometry, 0 Y=1"*() (publicHuman keys methods activities of encryption, y = x +xy2 encodingteachingdesignsecurebasic combinatorial oftransactions, andof plane compact learning flights, discs.stuctures.communications. shortest etc) paths. e3x 1 2 MANIPULATINGGilbert COMBINATORIAL Labelle / LaCIM - UQAM STRUCTURES Example 2.4A rooted tree on 1.7..kbc,..x,rt+3 finiteIn Combinatorics, construction amade structure,on a finites, is aset U. e 4 We say orthat : Us isIsis (built)equippedthe underlying on U with (theset of structure)s s Example 3.6An1,4 Iorienteda. 3 cyclea on IIf , 93 Exampleor 1.: s isA labelledtree on tJ by = thea, b,set c, U, + (4...;.ASP'kcL-.17 3 Variants:oddeven (oriented) (oriented) cycle cycle el" kit Example 4. (orA subsetbicoloration) on (of)tro 1%.ff., "1:\6 cif J R a y 0 _L0 Example 5.A (the) set structure on 3 Example 8.A partition= 10,*, 42,3, on 4, (7, '1'3 4 U /,0AAJ-1:1TJJ1Variants:oddeven set set Example 6. A permutation on ys..,.ITt s LY-=..egktN,Ii2/3000 /6, ai.".`t bd Example 9. An endofunction on f \--ii.el ;U ' 62-3 Se34 t a ,., ..* . 11.-1N 4 r g, ' e )% /I; . /a \--4 6 i --->.... f V241g13 e 1- 2} NN SPOf Example 7. Alinear order on U=1=d Al; .40;a3 eafj RioIf far.v.../6ri9 Vr..v,.631SyaS . H.2( er r17.-43"4iv2.,,/4itvy Iti, 'IVSoc.47,.. I 17ts 0,62, 44, F t 3 \,...0.,.........0...4".s:-",3..,..-er-'-ere...? a 3 xs44. 2 N*3 5 Example 10. ATJ=14,12,c,d,e, bicolored polygon f3on 5 Example 13. A singleton on U= I Al 6 (ThereExample is no14. singletonAn empty structurea set structure on U if (r..7(*on U=16 1.) Example 11. A graph on ET= it, .. .,ZSfq 4: /3,,-r,;--,-:---,24,..........___0#077.t.,ir 1 1 N, (There is no empty set structure if T7* +) ....,,,,,,.......4T....Ie.'4.....111-:,..... -.. 1 VAtei...---ip 23 2 99g Example 15.An Involution on U=1*-,4., A. , vS- 2..,a., qr,f5' h h A r rn s Example 12.ft' A connectedi9n graph on U= a. e 4 7 ,00.414/6 3 "111' ETC, ETC, ETC,... 7 For example, we can consider 8 CONVENTIONwill sometimes : A.beAn on represented arbitrary T...7 structureby thethethe speciesspeciesspeciesA ce.aof all evencycles,rootedtrees cycles,(arbres), trees(arbore5cences), b c thethethe speciesspeciesspecies E Ewa*vana)Eof allallall sets evensubsetsoddpermutations, (ensembles), sets, sets, (of sets), Aif LTstructure containsHere, A. U an on= even5 T..7Ad(odd)6., is c.,said ,A)number toe. be evenof elements. (odd) thethethe species speciesspeciesspecies End SL.VI'B ofof allallall edofunctions, linearbicoloredpartitions, orders, polygons) Twoif their structures underlying 4, sets and are t aredisjoint.said to be disjoint thethe species speciesspeciesempty 214.0,.species)(I of allthe0 graphs,connectedsingletons, empty(containing set, graphs, no structure), collectingStructures them can intobe studied SPECIESandOF classified STRUCTURES by : thethe species InvF ofof allallinvolutions,F- structures. 9 10 ByFLU] convention,:= the setfor anyof all finiteF- setstructures tJ we write on U PROPERTYtheFundamental proceeding 1. properties exampleFor every : satisfied finite set by t7, the set Example.thenIf F = = the species of cycles, PROPERTY 2.47E73EveryPVU bijectionIs= always rn, rijfinite. Is a 41- structure inducesCLU) another bijection,V = 14-111 ha c, 43I andCif ifrn,"if)1, 0)131= rn,n,1,, 11, then 0,1=144> 446 I411,24." "&tlivonANDirO;:74.."3Ar't 3gere't44417144,5r2 towhich form relabels a corresponding via /3 each C C-structure -structure ononV. U 0 Note. Two successive relabellingsvia A and Al11 CONVENTION. An arbitrary F -structure AL onT..712 amountseP4.)flop to a singlecmctaviou,relabelling )=CLA."3mei° via Cfpl A'o /3 : can be representedor by diagrams
Load more

Recommended publications
  • New Combinatorial Computational Methods Arising from Pseudo-Singletons Gilbert Labelle
    New combinatorial computational methods arising from pseudo-singletons Gilbert Labelle To cite this version: Gilbert Labelle. New combinatorial computational methods arising from pseudo-singletons. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.247-258. hal-01185187 HAL Id: hal-01185187 https://hal.inria.fr/hal-01185187 Submitted on 19 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2008, Valparaiso-Vi~nadel Mar, Chile DMTCS proc. AJ, 2008, 247{258 New combinatorial computational methods arising from pseudo-singletons Gilbert Labelley Laboratoire de Combinatoire et d'Informatique Math´ematique,Universit´edu Qu´ebec `aMontr´eal,CP 8888, Succ. Centre-ville, Montr´eal(QC) Canada H3C3P8 Abstract. Since singletons are the connected sets, the species X of singletons can be considered as the combinatorial logarithm of the species E(X) of finite sets. In a previous work, we introduced the (rational) species Xb of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that E(X) = exp(Xb) in the context of rational species, where exp(T ) denotes the classical analytical power series for the exponential function in the variable T .
    [Show full text]
  • An Introduction to Combinatorial Species
    An Introduction to Combinatorial Species Ira M. Gessel Department of Mathematics Brandeis University Summer School on Algebraic Combinatorics Korea Institute for Advanced Study Seoul, Korea June 14, 2016 The main reference for the theory of combinatorial species is the book Combinatorial Species and Tree-Like Structures by François Bergeron, Gilbert Labelle, and Pierre Leroux. What are combinatorial species? The theory of combinatorial species, introduced by André Joyal in 1980, is a method for counting labeled structures, such as graphs. What are combinatorial species? The theory of combinatorial species, introduced by André Joyal in 1980, is a method for counting labeled structures, such as graphs. The main reference for the theory of combinatorial species is the book Combinatorial Species and Tree-Like Structures by François Bergeron, Gilbert Labelle, and Pierre Leroux. If a structure has label set A and we have a bijection f : A B then we can replace each label a A with its image f (b) in!B. 2 1 c 1 c 7! 2 2 a a 7! 3 b 3 7! b More interestingly, it allows us to count unlabeled versions of labeled structures (unlabeled structures). If we have a bijection A A then we also get a bijection from the set of structures with! label set A to itself, so we have an action of the symmetric group on A acting on these structures. The orbits of these structures are the unlabeled structures. What are species good for? The theory of species allows us to count labeled structures, using exponential generating functions. What are species good for? The theory of species allows us to count labeled structures, using exponential generating functions.
    [Show full text]
  • A Logical Approach to Asymptotic Combinatorics I. First Order Properties
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATI-EMATlCS 65, 65-96 (1987) A Logical Approach to Asymptotic Combinatorics I. First Order Properties KEVIN J. COMPTON ’ Wesleyan University, Middletown, Connecticut 06457 INTRODUCTION We shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics; this framework provides methods for determining the probability that a large, finite structure, ran- domly chosen from a given class, will have a given property. Our main concern is the asymptotic probability: the limiting value as the size of the structure increases. For example, a common problem in elementary probability texts is to show that the asymptotic probabity that a per- mutation will have no fixed point is l/e. We shall say nothing about the closely related problem of determining rates of convergence, although the methods presented here may extend to such problems. To develop a general approach we must fix a language for specifying properties of structures. Thus, our approach is logical; logic is the branch of mathematics that deals with problems of language. In this paper we con- sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties. In the sequel to this paper we consider properties expressible in the more general language of monadic second order logic. Clearly, we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant. Therefore, we choose to consider only classes closed under disjoint unions and components (see Sect.
    [Show full text]
  • Interactions of Computational Complexity Theory and Mathematics
    Interactions of Computational Complexity Theory and Mathematics Avi Wigderson October 22, 2017 Abstract [This paper is a (self contained) chapter in a new book on computational complexity theory, called Mathematics and Computation, whose draft is available at https://www.math.ias.edu/avi/book]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: • Number Theory: Primality testing • Combinatorial Geometry: Point-line incidences • Operator Theory: The Kadison-Singer problem • Metric Geometry: Distortion of embeddings • Group Theory: Generation and random generation • Statistical Physics: Monte-Carlo Markov chains • Analysis and Probability: Noise stability • Lattice Theory: Short vectors • Invariant Theory: Actions on matrix tuples 1 1 introduction The Theory of Computation (ToC) lays out the mathematical foundations of computer science. I am often asked if ToC is a branch of Mathematics, or of Computer Science. The answer is easy: it is clearly both (and in fact, much more). Ever since Turing's 1936 definition of the Turing machine, we have had a formal mathematical model of computation that enables the rigorous mathematical study of computational tasks, algorithms to solve them, and the resources these require.
    [Show full text]
  • Middleware and Application Management Architecture
    Middleware and Application Management Architecture vorgelegt vom Diplom-Informatiker Sven van der Meer aus Berlin von der Fakultät IV – Elektrotechnik und Informatik der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften - Dr.-Ing. - genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Gorlatch Berichter: Prof. Dr. Dr. h.c. Radu Popescu-Zeletin Berichter: Prof. Dr. Kurt Geihs Tag der wissenschaftlichen Aussprache: 25.09.2002 Berlin 2002 D 83 Middleware and Application Management Architecture Sven van der Meer Berlin 2002 Preface Abstract This thesis describes a new approach for the integrated management of distributed networks, services, and applications. The main objective of this approach is the realization of software, systems, and services that address composability, scalability, reliability, and robustness as well as autonomous self-adaptation. It focuses on middleware for management, control, and use of fully distributed resources. The term integra- tion refers to the task of unifying existing instrumentations for middleware and management, not their replacement. The rationale of this work stems from the fact, that the current situation of middleware and management systems can be described with the term interworking. The actually needed integration of management and middleware concepts is still an open issue. However, identified trends in communications and computing demand for integrated concepts rather than the definition of new interworking scenarios. Distributed applications need to be prepared to be used and operated in a stable, secure, and efficient way. Middleware and service platforms are employed to solve this task. To guarantee this objective for a long-time operation, the systems needs to be controlled, administered, and maintained in its entirety, supporting the general aim of the system, and for each individual component, to ensure that each part of the system functions perfectly.
    [Show full text]
  • An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
    STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical
    [Show full text]
  • Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary
    Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its appli- cations. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its application-friendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators. The intellectual merits of this project include the development of applications of al- gebraic geometry and of combinatorial and combinatorial aspects of algebraic geometry. Specifically, Sottile will work to develop the theory and properties of orbitopes from the perspective of convex algebraic geometry, continue to investigate linear precision in geo- metric modeling, and apply the quantum Schubert calculus to linear systems theory.
    [Show full text]
  • Combinatorics and Graph Theory I (Math 688)
    Combinatorics and Graph Theory I (Math 688). Problems and Solutions. May 17, 2006 PREFACE Most of the problems in this document are the problems suggested as home- work in a graduate course Combinatorics and Graph Theory I (Math 688) taught by me at the University of Delaware in Fall, 2000. Later I added several more problems and solutions. Most of the solutions were prepared by me, but some are based on the ones given by students from the class, and from subsequent classes. I tried to acknowledge all those, and I apologize in advance if I missed someone. The course focused on Enumeration and Matching Theory. Many problems related to enumeration are taken from a 1984-85 course given by Herbert S. Wilf at the University of Pennsylvania. I would like to thank all students who took the course. Their friendly criti- cism and questions motivated some of the problems. I am especially thankful to Frank Fiedler and David Kravitz. To Frank – for sharing with me LaTeX files of his solutions and allowing me to use them. I did it several times. To David – for the technical help with the preparation of this version. Hopefully all solutions included in this document are correct, but the whole set is by no means polished. I will appreciate all comments. Please send them to [email protected]. – Felix Lazebnik 1 Problem 1. In how many 4–digit numbers abcd (a, b, c, d are the digits, a 6= 0) (i) a < b < c < d? (ii) a > b > c > d? Solution. (i) Notice that there exists a bijection between the set of our numbers and the set of all 4–subsets of the set {1, 2,..., 9}.
    [Show full text]
  • Combinatorial Species and Labelled Structures Brent Yorgey University of Pennsylvania, [email protected]
    University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 1-1-2014 Combinatorial Species and Labelled Structures Brent Yorgey University of Pennsylvania, [email protected] Follow this and additional works at: http://repository.upenn.edu/edissertations Part of the Computer Sciences Commons, and the Mathematics Commons Recommended Citation Yorgey, Brent, "Combinatorial Species and Labelled Structures" (2014). Publicly Accessible Penn Dissertations. 1512. http://repository.upenn.edu/edissertations/1512 This paper is posted at ScholarlyCommons. http://repository.upenn.edu/edissertations/1512 For more information, please contact [email protected]. Combinatorial Species and Labelled Structures Abstract The theory of combinatorial species was developed in the 1980s as part of the mathematical subfield of enumerative combinatorics, unifying and putting on a firmer theoretical basis a collection of techniques centered around generating functions. The theory of algebraic data types was developed, around the same time, in functional programming languages such as Hope and Miranda, and is still used today in languages such as Haskell, the ML family, and Scala. Despite their disparate origins, the two theories have striking similarities. In particular, both constitute algebraic frameworks in which to construct structures of interest. Though the similarity has not gone unnoticed, a link between combinatorial species and algebraic data types has never been systematically explored. This dissertation lays the theoretical groundwork for a precise—and, hopefully, useful—bridge bewteen the two theories. One of the key contributions is to port the theory of species from a classical, untyped set theory to a constructive type theory. This porting process is nontrivial, and involves fundamental issues related to equality and finiteness; the recently developed homotopy type theory is put to good use formalizing these issues in a satisfactory way.
    [Show full text]
  • Combinatorics: a First Encounter
    Combinatorics: a first encounter Darij Grinberg Thursday 10th January, 2019at 9:29pm (unfinished draft!) Contents 1. Preface1 1.1. Acknowledgments . .4 2. What is combinatorics?4 2.1. Notations and conventions . .4 1. Preface These notes (which are work in progress and will remain so for the foreseeable fu- ture) are meant as an introduction to combinatorics – the mathematical discipline that studies finite sets (roughly speaking). When finished, they will cover topics such as binomial coefficients, the principles of enumeration, permutations, parti- tions and graphs. The emphasis falls on enumerative combinatorics, meaning the art of computing sizes of finite sets (“counting”), and graph theory. I have tried to keep the presentation as self-contained and elementary as possible. The reader is assumed to be familiar with some basics such as induction proofs, equivalence relations and summation signs, as well as have some experience with mathematical proofs. One of the best places to catch up on these basics and to gain said experience is the MIT lecture notes [LeLeMe16] (particularly their first five chapters). Two other resources to familiarize oneself with proofs are [Hammac15] and [Day16]. Generally, most good books about “reading and writing mathemat- ics” or “introductions to abstract mathematics” should convey these skills, although the extent to which they actually do so may differ. These notes are accompanying two classes on combinatorics (Math 4707 and 4990) I am giving at the University of Minneapolis in Fall 2017. Here is a (subjective and somewhat random) list of recommended texts on the kinds of combinatorics that will be considered in these notes: • Enumerative combinatorics (aka counting): 1 Notes on graph theory (Thursday 10th January, 2019, 9:29pm) page 2 – The very basics of the subject can be found in [LeLeMe16, Chapters 14– 15].
    [Show full text]
  • A COMBINATORIAL THEORY of FORMAL SERIES Preface in His
    A COMBINATORIAL THEORY OF FORMAL SERIES ANDRÉ JOYAL Translation and commentary by BRENT A. YORGEY This is an unofficial translation of an article that appeared in an Elsevier publication. Elsevier has not endorsed this translation. See https://www.elsevier.com/about/our-business/policies/ open-access-licenses/elsevier-user-license. André Joyal. Une théorie combinatoire des séries formelles. Advances in mathematics, 42 (1):1–82, 1981 Preface In his classic 1981 paper Une théorie combinatoire des séries formelles (A com- binatorial theory of formal series), Joyal introduces the notion of (combinatorial) species, which has had a wide-ranging influence on combinatorics. During the course of researching my PhD thesis on the intersection of combinatorial species and pro- gramming languages, I read a lot of secondary literature on species, but not Joyal’s original paper—it is written in French, which I do not read. I didn’t think this would be a great loss, since I supposed the material in his paper would be well- covered elsewhere, for example in the textbook by Bergeron et al. [1998] (which thankfully has been translated into English). However, I eventually came across some questions to which I could not find answers, only to be told that the answers were already in Joyal’s paper [Trimble, 2014]. Somewhat reluctantly, I found a copy and began trying to read it, whereupon I discovered two surprising things. First, armed with a dictionary and Google Translate, reading mathematical French is not too difficult (even for someone who does not know any French!)— though it certainly helps if one already understands the mathematics.
    [Show full text]
  • Analytic Combinatorics for Computing Seeding Probabilities
    algorithms Article Analytic Combinatorics for Computing Seeding Probabilities Guillaume J. Filion 1,2 ID 1 Center for Genomic Regulation (CRG), The Barcelona Institute of Science and Technology, Dr. Aiguader 88, Barcelona 08003, Spain; guillaume.fi[email protected] or guillaume.fi[email protected]; Tel.: +34-933-160-142 2 University Pompeu Fabra, Dr. Aiguader 80, Barcelona 08003, Spain Received: 12 November 2017; Accepted: 8 January 2018; Published: 10 January 2018 Abstract: Seeding heuristics are the most widely used strategies to speed up sequence alignment in bioinformatics. Such strategies are most successful if they are calibrated, so that the speed-versus-accuracy trade-off can be properly tuned. In the widely used case of read mapping, it has been so far impossible to predict the success rate of competing seeding strategies for lack of a theoretical framework. Here, we present an approach to estimate such quantities based on the theory of analytic combinatorics. The strategy is to specify a combinatorial construction of reads where the seeding heuristic fails, translate this specification into a generating function using formal rules, and finally extract the probabilities of interest from the singularities of the generating function. The generating function can also be used to set up a simple recurrence to compute the probabilities with greater precision. We use this approach to construct simple estimators of the success rate of the seeding heuristic under different types of sequencing errors, and we show that the estimates are accurate in practical situations. More generally, this work shows novel strategies based on analytic combinatorics to compute probabilities of interest in bioinformatics.
    [Show full text]