Weak separation, positivity and extremal Yangian invariants

We classify all positive n -particle N k MHV Yangian invariants in 𝒩 = 4 YangMills theory with n = 5 k , which we call extremal because none exist for n > 5 k . 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. 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.