# CSM

## Seminar

### Introduction to species and combinatorial equations

Seminar Room 1, Newton Institute

#### Abstract

1) Elementary introduction to species through examples : Collecting structures into species leading to the general definition of a combinatorial species. Exponential generating series of a species for labelled enumeration. Elementary combinatorial operations and equations between species.

2) More advanced theory : Cycle index and tilda series of a species for unlabelled enumeration. Equipotence versus combinatorial equality. Taking connected components. Combinatorial equations for weighted connected and 2-connected graphs. Explicit examples.

*Note : This lecture is preparatory to the reading of papers of Pierre Leroux and collaborators. These papers are available below. Other material, including a link to the book "Combinatorial Species and Tree-like Structures", can be found on Pierre Leroux's web page

Introduction to the Theory of Species of Structures, by François Bergeron, Gilbert Labelle, and Pierre Leroux.

Papers and slides on Mayer graph weights:

- Enumerative problems inspired by Mayer's theory of cluster integrals, by Pierre Leroux.
- Graph weights arising from Mayer's theory of cluster integrals, slides by Pierre Leroux.
- Graph weights arising from Mayer's theory of cluster integrals, by G. Labelle, P. Leroux and M. G. Ducharme.
- Note on Legendre transform and line-irreducible graphs, by David Brydges and Pierre Leroux.

Papers and slides on two-connected graphs:

- 2-connected graphs with prescribed three-connected components, slides by Pierre Leroux.
- 2-connected graphs with prescribed 3-connected components, slides by Pierre Leroux.
- Two-connected graphs with prescribed three-connected components, by Andrei Gagarin, Gilbert Labelle, Pierre Leroux and Timothy Walsh.

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