Grass trees and forests: Enumeration of Grassmannian trees and forests, with applications to the momentum amplituhedron

The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher's result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumerate bipartite planar trees and forests, and we apply our results to enumerate various families of permutations: for example, bipartite planar trees are in bijection with separable permutations. Our generating function for Grassmannian forests can be interpreted as the rank generating function for the boundary strata of the momentum amplituhedron, an object which encodes the tree-level S-matrix of maximally supersymmetric Yang-Mills theory. This allows us to verify that the Euler characteristic of the momentum amplituhedron is 1.