Colored operads, series on colored operads, and combinatorial generating systems

A new sort of combinatorial generating system, called bud generating system, is introduced. Bud generating systems are devices for specifying sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing to work with all these generating systems in a unified way. The theory of bud generating systems presented here heavily uses the one of colored operads. Indeed, an object is generated by a bud generating system if it satisfies a certain equation in a colored operad. Moreover, with the aim to compute the generating series of the languages of bud generating systems, we introduce formal power series on colored operads and several operations on these: a pre-Lie product, an associative product, and two analogues of the Kleene star operation. Series on colored operads intervene to express in several ways the languages specified by bud generating systems and allow to enumerate combinatorial objects with respect to some statistics. Some examples of bud generating systems are constructed, in particular to specify some sorts of balanced trees and specific intervals in the Tamari lattices, and obtain recursive formulae enumerating these. CONTENTS Introduction 2 1. Colored operads and bud operads 7 1.1. Colored operads 7 1.2. Free colored operads 9 1.3. Bud operads 12 2. Formal power series on colored operads 14 2.1. The space of series on colored operads 14 2.2. Pre-Lie product on series 18 2.3. Composition product on series 23 3. Bud generating systems and combinatorial generation 29 3.1. Bud generating systems 29 3.2. First properties 30 3.3. Links with other generating systems 31 4. Series on colored operads and bud generating systems 34 4.1. Hook generating series 35 4.2. Syntactic generating series 38 4.3. Synchronous generating series 43 5. Examples 47 5.1. Monochrome operads and bud operads 47 5.2. Series on colored operads 49 5.3. Bud generating systems 51 5.4. Series of bud generating systems 57 Conclusion and perspectives 64 References 65 Date: May 17, 2016. 2010 Mathematics Subject Classification. 05C05, 18D50, 68Q42, 32A05.