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

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.Comment: 48 page

Generating functions for generating trees

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and published in its vol. 246(1-3), March 2002, pp. 29-5

Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations

We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the constructible classes of Flajolet and Sedgewick with Joyal's species theory. We extend the implicit species theorem to structures of size zero. A quadratic iterative Newton method is shown to solve well-founded systems combinatorially. From there, truncations of the corresponding generating series are obtained in quasi-optimal complexity. This iteration transfers to a numerical scheme that converges unconditionally to the values of the generating series inside their disk of convergence. These results provide important subroutines in random generation. Finally, the approach is extended to combinatorial differential systems.Comment: 61 page

Noncommutative Symmetric Systems over Associative Algebras

This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it ${\mathcal N}$CS $(\text{noncommutative symmetric})$ systems} over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$CS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion {\it ${\mathcal N}$CS systems} over associative $\mathbb Q$-algebras. We then prove some results for ${\mathcal N}$CS systems in general; the ${\mathcal N}$CS systems over bialgebras or Hopf algebras; and the universal ${\mathcal N}$CS system formed by the generating functions of certain NCSFs in [GKLLRT]. Finally, we review some of the main results that will be proved in the followed papers [Z4], [Z6] and [Z7] as some supporting examples for the general discussions given in this paper.Comment: A connection of NCS systems with combinatorial Hopf algebras of M. Aguiar, N. Bergeron and F. Sottile has been added in Remark 2.17. Latex, 32 page