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.

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.