# Weak Separation, Positivity and Extremal Yangian Invariants

## TL;DR

This paper classifies all extremal positive Yangian invariants in N=4 super Yang-Mills theory for specific particle numbers, linking the problem to enumerating plane cactus graphs with pentagons, and provides explicit formulas and classifications.

## Contribution

It introduces a complete classification of extremal positive Yangian invariants for n=5k, connecting the problem to plane cactus graphs and weak separation, with explicit enumeration formulas.

## Key findings

- Classified all extremal positive Yangian invariants for n=5k.
- Established a correspondence with plane cactus graphs with k pentagons.
- Provided explicit formulas and a simple rule for constructing these invariants.

## Abstract

We classify all positive n-particle N^kMHV Yangian invariants in N=4 Yang-Mills theory with n=5k, which we call extremal because none exist for n>5k. We show that this problem is equivalent to that of enumerating plane cactus graphs with k pentagons. We use the known solution of that problem to provide an exact expression for the number of cyclic classes of such invariants for any k, and a simple rule for writing them down explicitly. As a byproduct, we provide an alternative (but equivalent) classification by showing that a product of k five-brackets with disjoint sets of indices is a positive Yangian invariant if and only if the sets are all weakly separated.