Nonsymmetric operads in combinatorics Samuele Giraudo Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEM, F-77454, Marne-la-Vallée, France Email address [email protected] : arXiv:2104.12398v1 [math.CO] 26 Apr 2021 Mathematics Subject Classification. 2010 05-00, 05C05, 05E15, 16T05, 18D50. Key words and phrases. binatorics; Algebraic combinatorics; Computer science; Com Tree; Formal power series; Rewrite system; Operad; Bialgebra; Hopf bialgebra; Pre-Lie algebra; Dendriform algebra. Operads are algebraic devices offering a formalization of the Abstract. concept of operations with several inputs and one output. Such opera- tions can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompany- ing it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide con- nections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their proper- ties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this the- ory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combina- torics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmet- ric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics. April 27, 2021 Contents Introduction 1 Chapter 1. Enriched collections 9 1. Collections 9 1.1. Structured collections 9 1.2. Operations over collections 12 1.3. Examples 19 2. Posets 23 2.1. Posets on collections 23 2.2. Examples 24 3. Rewrite systems 25 3.1. Rewrite systems on collections 26 3.2. Examples 28 Chapter 2. Treelike structures 33 1. Planar rooted trees 33 1.1. Collection of planar rooted trees 33 1.2. Subcollections of planar rooted trees 37 2. Syntax trees 40 2.1. Collections of syntax trees 40 2.2. Grafting operations 42 2.3. Patterns and rewrite systems 44 3. Treelike structures 48 3.1. Rooted trees 49 3.2. Colored syntax trees 49 Chapter 3. Algebraic structures 53 1. Polynomials spaces 53 1.1. Polynomials on collections 53 1.2. Operations over polynomial spaces 56 iii iv CONTENTS 1.3. Changes of basis and posets 59 2. Bialgebras 61 2.1. Biproducts on polynomial spaces 61 2.2. Products on polynomial spaces 63 2.3. Polynomial bialgebras 66 3. Types of polynomial bialgebras 69 3.1. Associative and coassociative algebras 69 3.2. Dendriform algebras 71 3.3. Pre-Lie algebras 72 3.4. Hopf bialgebras 73 Chapter 4. Nonsymmetric operads 81 1. Operads as polynomial spaces 81 1.1. Composition maps 81 1.2. Operads 87 1.3. Algebras over operads 89 2. Free operads, presentations, and Koszulity 92 2.1. Free operads 92 2.2. Presentations by generators and relations 93 2.3. Koszulity 95 3. Main operads 97 3.1. Operads of words 97 3.2. Operads of trees 102 3.3. Operads of graphs 107 Chapter 5. Applications and generalizations 117 1. Series on operads 117 1.1. Series on algebraic structures 117 1.2. Generalizing series multiplication 119 1.3. Generalizing series composition 121 2. Enriched operads 123 2.1. Colored operads 123 2.2. Cyclic operads 126 2.3. Symmetric operads 128 3. Product categories 132 3.1. Abstract bioperators 132 CONTENTS v 3.2. Pros 134 3.3. Main pros 136 Bibliography 141 Introduction Combinatorics, algebra, and programming Let us start this text by considering two fields of mathematics: combinatorics and algebra, and a field of computer science: programming. These topics have far more in common than it appears at first glance and we shall explain why. Combinatorics. One of the most common activities in combinatorics consists in studying families of combinatorial objects such as permutations, words, trees, or graphs. One of the most common activities about such families is to provide formulas to enu- merate them according to a decent size notion. A large number of tools exists for this purpose such as manipulation of generating series, use of bijections, and use of decom- positions rules to break objects into elementary pieces. This last tool is crucial and relies on the fact that if one knows a way to compose objects to build a bigger one, then with- out much effort one knows a way to decompose them. Let us, for instance, consider the celebrated set of Motzkin paths. Recall that a Motzkin path is a walk in the quarter plane connecting the origin to a point on the x-axis by means of steps , , and . Let us denote by u ; u the binary composition rule consisting in substituting u (resp. u ) 1 2) 1 2 with the first( (resp. second) point of , and by u ; u ; u the ternary composition 1 2 3 rule consisting in substituting u (resp. u , u ) with the( first (resp.) second, third) point of 1 2 3 . By using these two operations, one can, for instance, provide the decomposition ; ; ; ; ; (0.0.1) = ( ) of the Motzkin path appearing as left member. An easy result says that a nonempty Motzkin path decomposes in a unique way either as a step followed by a Motzkin path, or as a step followed by a Motzkin path, a step , and another Motzkin path. This leads to the expression M { } ⊔ { ; u u ∈ M} ⊔ { ; u ; u u ; u ∈ M} (0.0.2) = ( 2) : 2 ( 2 3) : 2 3 describing the set M of all Motzkin paths. From (0.0.2), one harvests combinatorial data as a formula to count Motzkin paths of a given size (with respect to an adequate notion of size), bijections between Motzkin paths and other families of combinatorial objects admitting similar decompositions (like Motzkin trees or Motzkin configurations 1 2 INTRODUCTION of chords in polygons), or even the definition of generalizations of these objects (by adding colors on the steps, or by adding other possible sorts of steps). Algebra. On another side, in algebra, the study of algebraic structures consisting in a set of operations and the relations they satisfy is very habitual. To specify such an algebraic structure, one provides the nontrivial relations satisfied by its operations. For instance, any binary operation ⋆ satisfying for any inputs x , x , and x the relation 1 2 3 x ⋆x ⋆x − x ⋆ x ⋆x x ⋆x ⋆x − x ⋆ x ⋆x (0.0.3) ( 1 2) 3 1 ( 2 3) = ( 1 3) 2 1 ( 3 2) specifies the class of the so-called pre-Lie algebras. Classical examples include among others monoids, groups, lattices, associative algebras, commutative algebras, and dendri- form algebras. An obvious but important fact is that expressions and relations can be composed: the variables in a relation refer to any term of the algebraic structure. For instance, in (0.0.3), all the occurrences of x can be replaced by a term x4 ⋆ x5, leading to 2 the relation x ⋆ x4 ⋆ x5 ⋆x − x ⋆ x4 ⋆ x5 ⋆x x ⋆x ⋆ x4 ⋆ x5 − x ⋆ x ⋆ x4 ⋆ x5 (0.0.4) ( 1 ( )) 3 1 (( ) 3) = ( 1 3) ( ) 1 ( 3 ( )) that still holds. Furthermore, by seeing the binary operation ⋆ as a device ⋆ (0.0.5) x x 1 2 having two inputs x and x and one output x ⋆x , Relation (0.0.3) reformulates as 1 2 1 2 ⋆ ⋆ ⋆ ⋆ − − ; (0.0.6) ⋆ x x ⋆ ⋆ x x ⋆ 3 1 = 2 1 x x x x x x x x 1 2 2 3 1 3 3 2 and Relation (0.0.4) as ⋆ ⋆ ⋆ ⋆ ⋆ x x ⋆ x ⋆ 3 − 1 ⋆ ⋆ − 1 : (0.0.7) x ⋆ ⋆ x = x ⋆ 1 3 3 x x x x x x x4 x5 x x 4 5 4 5 1 3 4 5 Therefore, one observes that algebraic expressions translates as syntax trees, and that composition and substitution of expressions translates as grafting of trees. Programming. In programming, and more precisely in the context of functional programming, one encounters expressions of the form v1; v2 ; g2 v3 ; v4 (0.0.8) f(g1( ) ( ) ) where f, g1, and g2 are function names, and v1, v2, v3, and v4 are value names. This func- tion call can be considered as a program, and the returned value is the value computed by the execution of the program. When our programming language satisfies referential NONSYMMETRIC OPERADS AS A MEETING POINT 3 transparency (that is, any expression can be substituted by its value without changing the overall computation), the call (0.0.8) is equivalent to both the calls f w1; g2 v3 ; v4 and f g1 v1; v2 ; w2; v4 (0.0.9) ( ( ) ) ( ( ) ) where w1 is the value of g1 v1; v2 and w2 is the value of g2 v3 . As a consequence, this ( ) ( ) leads to the fact that the order of evaluation of the sub-expressions g1 v1; v2 and g2 v3 does not influence the computation of (0.0.8). Furthermore, by seeing( the functions) ( f1,) g1, and g2 as black boxes ; ; (0.0.10) f g1 g2 where inputs are depicted below and outputs above the boxes, the program (0.0.8) is made of compositions of such black boxes where inputs are completed with values (that are arguments of the function calls).