# Statistics on Graphs, Exponential Formula and Combinatorial Physics

## TL;DR

This paper develops a general framework for applying the exponential formula, a fundamental combinatorial identity, across various fields like physics, mathematics, and computer science, to analyze complex structures via their connected components.

## Contribution

It introduces a unified framework to systematically utilize the exponential formula in diverse contexts, enhancing understanding and application of combinatorial structures.

## Key findings

- Unified framework for exponential formula applications
- Broadened understanding of combinatorial structures
- Potential for cross-disciplinary applications

## Abstract

The concern of this paper is a famous combinatorial formula known under the name "exponential formula". It occurs quite naturally in many contexts (physics, mathematics, computer science). Roughly speaking, it expresses that the exponential generating function of a whole structure is equal to the exponential of those of connected substructures. Keeping this descriptive statement as a guideline, we develop a general framework to handle many different situations in which the exponential formula can be applied.