From 4gravitons' Post on the Amplitudes 2017 Conference: "Between the two of them, Nima (Arkani-Hamed) and Yuntao (Bai) covered an interesting development, tying the Amplituhedron together with the string theory-esque picture of scattering amplitudes pioneered by Freddy Cachazo, Song He, and Ellis Ye Yuan (or CHY). There’s a simpler (and older) Amplituhedron-like object called the associahedron that can be thought of as what the Amplituhedron looks like on the surface of a string, and CHY’s setup can be thought of as a sophisticated map that takes this object and turns it into the Amplituhedron."
The coarse h-polynomials of the associahedra (and their simplicial duals) are the Narayana polynomials A001263 while the refined h-partition polynomials are the noncrossing partitions A134264. The refined f-partition polynomials or face-partition polynomials of the associahedra are given by A133437 (unsigned and renormalized).
The permutohedra, whose h-polynomials are the Eulerian polynomials A008292 and f-partition polynomials A049019, are also discussed in (1) below.
Associated references are
(1) "Scattering forms and the positive geometry of kinematics, color, and the worldsheet" by N. Arkani-Hamed, Y. Bai, S. He, and G. Yan
(2) "Labelled tree graphs, Feynman diagrams and disk integrals" by X. Gao , S. He , Y. Zhang
(3) "Scattering from geometries" by S. He.
"The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint $\phi~^3$ in scalar theory, we establish a direct connection between its “scattering form” and a classic polytope—the associahedron—known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula."