# A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

## TL;DR

This paper presents a recursive formula to enumerate connected Feynman diagrams with any number of external legs in a fermionic gas, facilitating fast computations and asymptotic analysis.

## Contribution

It generalizes a recursive enumeration method for connected Feynman diagrams to arbitrary external legs and orders, based on combinatorial structures and Wick's theorem.

## Key findings

- Derived exact recurrence relations for diagrams with two and four external legs.
- Enabled efficient computation of diagram counts for large cases.
- Provided asymptotic expansion for the number of connected diagrams.

## Abstract

In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration is the Wick theorem, permitting an easy generalization to any quantum field theory. The iterative enumeration is constructive and enables a fast computation of the number of connected Feynman diagrams for a large amount of cases. In particular, the recurrence is solved exactly for two and four external legs, leading to the asymptotic expansion of the number of different connected Feynman diagrams.