Enumerative and bijective aspects of combinatorial maps: generalization, unification and application (PhD thesis)

TL;DR

This thesis explores the enumeration of combinatorial maps, their algebraic structures, and applications to counting various combinatorial objects, unifying different enumeration problems through maps and constellations.

Contribution

It introduces new enumeration methods for maps and constellations, unifies various combinatorial enumeration problems, and links maps to other combinatorial objects like lattices and graphs.

Findings

Enumeration formulas for maps and constellations

Unified framework for symmetric group factorizations

Connections between maps and other combinatorial structures

Abstract

This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumeration can also help us count other combinatorial objects. This thesis is a sampling from the rich results and connections in the enumeration of maps. This thesis is structured into four major parts. The first part, including Chapter 1 and 2, consist of an introduction to the enumerative study of maps. The second part, Chapter 3 and 4, contains my work in the enumeration of constellations, which are a special type of maps that can serve as a unifying model of some factorizations of the identity in the symmetric group. The third part, composed by Chapter 5 and 6, shows my research on the enumerative link from maps to other combinatorial objects, such as generalizations of the Tamari lattice and random graphs embeddable onto surfaces. The last part is the closing chapter, in which the thesis concludes with some perspectives and future directions in the enumerative study of maps.