Feynman Diagrams in Algebraic Combinatorics

TL;DR

This paper demonstrates how quantum field theory techniques can be rigorously applied to algebraic combinatorics, deriving formulas for power series reversion, composition, and inversion in a detailed, accessible manner.

Contribution

It introduces a categorified Faa di Bruno formula, explicit reversion formulas, and a proof of Lagrange-Good inversion using quantum field theory tools, bridging physics and combinatorics.

Findings

Derived a categorified Faa di Bruno formula for multivariable composition

Provided explicit formulas for power series reversion

Proved Lagrange-Good inversion in a rigorous framework

Abstract

We show, in great detail, how the perturbative tools of quantum field theory allow one to rigorously obtain: a ``categorified'' Faa di Bruno type formula for multiple composition, an explicit formula for reversion and a proof of Lagrange-Good inversion, all in the setting of multivariable power series. We took great pains to offer a self-contained presentation that, we hope, will provide any mathematician who wishes, an easy access to the wonderland of quantum field theory.