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

TL;DR

This paper introduces bud generating systems based on colored operads for unified combinatorial object generation, providing a framework for enumeration and recursive formulas across various grammar types.

Contribution

It develops a new formalism using colored operads to unify different combinatorial generating systems and introduces series on colored operads for enumeration.

Findings

Unified framework for context-free, regular, and synchronous grammars

Introduction of formal power series on colored operads for enumeration

Recursive formulas for balanced trees and other combinatorial objects

Abstract

We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing us to work with all these generating systems in a unified way. The theory of bud generating systems uses colored operads. Indeed, an object is generated by a bud generating system if it satisfies a certain equation in a colored operad. 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. Series on colored operads are crucial to express the languages specified by bud generating systems and allow us 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 to obtain recursive formulas enumerating these.