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

Abstract

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. Contents 1. Introduction 2. Variations of the Exponential Formula 3. Grassmannian graphs, trees and forests 4. Separable permutations and Grassmannian tree permutations 5. The Momentum Amplituhedron Appendix A. Data References

On the geometry of the orthogonal momentum amplituhedron

“…Applying this algorithm to the orthogonal momentum amplituhedron O k , we observe that all boundaries can be labelled by a particular class of graphs, which we call orthogonal Grassmannian forests, that correspond to all possible factorizations of ABJM amplitudes. This observation is analogous to the one that has been made for N = 4 SYM in [16], where the boundaries of the momentum amplihedron M n,k can be labelled using Grassmannian forests, see [17]. Both form a subset of the Grassmannian graphs introduced in [18].…”

“…Both form a subset of the Grassmannian graphs introduced in [18]. In this paper we summarise our explorations of the boundaries for O k for k ≤ 7, and provide a conjecture on the boundary stratification for all k. In particular, using the methods developed in [17], we derive a generating function for orthogonal Grassmannian forests. This conjecturally enumerates the boundaries of O k of a given dimension and allows us to argue that the Euler characteristic for the orthogonal momentum amplituhedron equals one.…”

“…An acyclic Grassmannian graph is called a Grassmannian forest while a connected acyclic Grassmannian graph is called a Grassmannian tree [17]. Given a Grassmanian forest F , we denote the set of Grassmannian trees in F by Trees(F ).…”

“…Following the definition for boundaries of the momentum amplituhedron given in [17], we define boundaries of the orthogonal momentum amplituhedron as follows: given an orthitroid cell S σ of OG + (k), we say that…”

“…On the other hand, even-particle Mandelstam variables correspond to factorization into two odd-particle scattering amplitudes. It is known that ABJM theory only JHEP12(2022)006 It was observed in [22], and later clarified in [17], that the boundaries of the momentum amplituhedron M n,k are in bijection with (contracted) Grassmannian forests of type (k, n). We conjecture that an analogous statement is true for the orthogonal momentum amplituhedron: boundaries of the orthogonal momentum amplituhedron O k are in bijection with OG forests of type k. More precisely, Φ • Γ is a boundary of O k if and only if Γ is an OG forest of type k. We have explicitly verified this characterization for the boundaries of O k for k ≤ 7.…”