Universita` degli Studi di Torino Scuola di Scienze della Natura Corso di Studi in Matematica Combinatorial species and transitive relations Tesi di laurea di secondo livello - A.A. 2012/2013 candidato: relatore: Daniele P. Morelli Prof.ssa Lea Terracini Combinatorial species and transitive relations Daniele P. Morelli Contents Preface 4 Introduction 8 1 Preliminary Notions 9 1.1 Integer sequences and power series . 9 1.2 Obtaining generating function from recurrences . 13 1.3 Labeled and unlabeled structures . 15 1.4 Basic notions about categories . 16 1.5 Species of Structures . 18 1.6 Series associated to species . 18 1.7 Isomorphism of species . 21 2 Classical enumerative tools 23 2.1 Stirling numbers of the second kind . 23 2.2 Burnside's lemma . 24 2.3 M¨obiusinversion . 26 2.4 Partitions . 26 2.5 Euler Transform . 29 3 Operations on species 31 3.1 Sum of species . 31 3.2 Product of species . 32 3.3 Composition of species . 36 3.4 Derivative of a species . 41 3.5 Pointing of a species . 43 3.6 Cartesian product of species . 44 3.7 Functorial composition . 45 4 Relation theory 47 4.1 Basic notions and results about relations . 47 4.2 Operations on relations . 49 4.3 Posets, order ideals and lattices . 50 2 5 Algebraic approaches to partial orders 53 5.1 Monomial orders and Gr¨obnerbases . 53 5.2 Toric ideals . 55 5.3 Hibi ideal of a poset . 57 5.4 Toric ideals associated to posets . 57 5.5 Order ideals and cartesian product of posets . 58 5.6 Adjoint functors in the category of posets . 60 6 Counting transitive relations 63 6.1 Transitivity . 63 6.2 Indecomposable partial orders . 67 6.3 Graded orders . 71 6.4 Finite topologies . 74 7 Conclusions 77 7.1 Experimental investigations . 77 7.2 Posets and graded posets . 79 Appendix 82 A1 Common species and operations . 82 A2 PARI/gp Routines . 84 A3 Tables of integer sequences . 89 References 96 3 Preface The use of power series for combinatorial purposes dates back at least to the age of Euler, who gave some of the first non-trivial examples of ordinary generating functions. Some times later Laplace had the insight that alge- braic operations on power series correspond to set-theoretic manipulations on the object counted by the sequences of coefficients of such series. As al- ways happens in mathematics, several decades of studies and investigations were needed to understand how a bunch of technical tools can in fact be unified in a general setting, and a theoretical background be given to such mathematical machinery. When a scientifical discipline needs clarification on a foundational level, the most problematic aspects appear to emerge with respect to definitions rather than theorems. For instance, a good definition of combinatorial struc- ture is not easy to formulate, and a lot of philosophical questions must be answered: what is the difference between labeled and unlabeled structures? Is it possible to formulate the concept of labeled structure without refer- ring explicitly to a specific set of labels but without making the structure unlabeled? Is it possible to give a rigorous combinatorial interpretation of functional composition? Is there a general notion of isomorphism, which is independent of the particular kind of structure considered? Is there a functional equivalent of the action of making a vertex distinguishable from the others (like in rooted trees)? How does the action of the symmetric group Sn over a set with n elements induces a correspondent action over the structures of some kind built over this set? The theory of combinatorial species, introduced by Andr´eJoyal in the Eighties (see [6]), constitutes an unifying step in the foundations of combi- natorics. It provides a general setting for most well-known results in discrete mathematics, and a solid background for the understanding and the use of generating functions and of the study of labeled and unlabeled combinatorial structures. It also helps solving most of the previously mentioned questions. This theory helps us understand most previously known but somewhat obscure facts, and to give new insights in the understanding of combina- torics. We could trace a parallel with Category theory (that in fact provides the basic notions on which the theory of species is constructed), which is a well-known example of a generalizing instrument that made clear a lot of known results. The concept of the fundamental group of a topological space, for instance, would be a lot less clear if we did not know what a category is and what a functor is. In the words of Gian-Carlo Rota:1 Every discovery of a new scientific fact is a challenge to uncover the uderlying mathematical structure. This structure is not \ab- 1From [11]. 4 stracted" from nature, as psychologists would have us believe. It is the basic makeup of nature, it was always there, waiting to be told and staring at us all the time. The natural laws discovered by scientists will be refined like a metal, polished like a jewel and finally stored as theorems in the archives of mathematics. Math- ematicians triumphantly point to mechanics as the example of a theory that began as an empirical science, and that eventually made its way into mathematics as a generalized geometry, geom- etry with time added. Mathematicians believe that every science will sooner or later meet the fate that befell mechanics. This may sound excessively positivist, but it is undeniable that the most important advances in mathematics are those that help us understand the \underlying mathematical structure" of some kind of discovery. Even the solution of a big problem is most often credited for the new technical ma- chinery provided and for the general understanding of a subject, rather than for supplying a mere answer to a mathematical question. On the other hand, we must admit that a good theory should in fact be able to solve new and important problems. We are confident that the theory of species will soon prove itself worth of this task, and this work would like to provide some insights that this theory allows us to make. We'd like to cite Rota again:2 It would probably be counterproductive to let it be known that behind every \genius" there lurks a beehive of research mathe- maticians who gradually built up to the “final” step in seemingly pointless research papers. And it would be fatal to let it be known that the showcase problems of mathematics are of little or no in- terest for the progress of mathematics. [...] There is a second way by which mathematics advances, one that mathematicians are also reluctant to publicize. It happens when- ever some commonsense notion that had heretofore been taken for granted is discovered to be wanting, to need clarification or definition. Such foundational advances produce substantial divi- dends, but not right away. I like to remember how seemingly small changes in mathematical nota- tion can drastically enhance the power of our thinking. The first use of the arrow notation f : X ! Y for the identification of a function dates around 1941.3 Before that date, the notation f(X) ⊆ Y was mostly used. Very few years after, the theory of categories was born. 2From the foreword of [1]. 3W. Hurewicz, On duality theorems, Bull. Am. Math. Soc., 47, 562{563 (1941), as cited in [9]. 5 This is not just an historical curiosity. Category theory is the discipline of arrows, after all. Obviously, it could have been discovered even before the introduction of the arrow notation, but it didn't, because a bunch of symbols were needed to let us keep in mind that, verily, a function is a map. The theory of species seems to be something like that: a new way to define things and refer to them. It is useful because most important com- binatorial constructions are automatically handled by the algebraic tools provided by it, and it is enlightning because it gives an elegant theoretical instrumentation that also helps us recall the true (onthological and ethimo- logical) nature of combinatorics: not simply the art of counting, but the art of combining. 6 7 Introduction This work is intended to introduce the theory of combinatorial species and to give some ideas about how it may help to find solutions to some open problems in enumerative combinatorics. In particular, we focused on the enumeration of binary relations over a finite set, specifically transitive rela- tions. The first section is intended to review some basic facts about combina- torics and to introduce the notion of combinatorial species. Basic notions about integer sequences and power series are recalled, and standard ways to relate those sequences to power series are introduced. Fundamental re- sults about ordinary and exponential generating functions are stated, and a general notion of isomorphism is defined, leading to the dichotomy between labeled and unlabeled structures. Section two is devoted to the study of some well-known enumerative tools which will be useful in the foregoing discussion. Stirling numbers of the second kind, Burnside's lemma and the basic theory of integer partitions (including Euler transform) are treated. Also, a brief introduction to M¨obius inversion is given. The third section is a survey of the operations that can be defined over combinatorial species. Some classical examples are introduced and we'll state and prove very important results relating those operations to functional operations over the corresponding generating series. Section four introduces the theory of binary relations, with particular attention to the transitive ones. Some important results about posets are recalled (including a celebrated result by Garrett Birkhoff).