What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory?


(This is a severely reduced version of the original post, which can be found at my blog <https://tcjpn.wordpress.com/2016/11/20/an-intriguing-tapestry-number-triangles-polytopes-grassmannians-and-scattering-amplitudes/>, with numerous references to the combinatorics of the associahedra and permutohedra and relations to scattering theory and Grassmannians.)



An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
From 4gravitons' [Post on the Amplitudes 2017 Conference] [1](https://4gravitons.wordpress.com/2017/07/20/more-travel/): "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](https://oeis.org/A001263) while the refined h-partition polynomials are the noncrossing partitions [A134264](https://oeis.org/A134264). The refined f-partition polynomials or face-partition polynomials of the associahedra are given by [A133437](https://oeis.org/A133437) (unsigned and renormalized).


The permutohedra, whose h-polynomials are the Eulerian polynomials [A008292](https://oeis.org/A008292) and f-partition polynomials [A049019](https://oeis.org/A049019), are also discussed in (1) below.


Associated references are


(1) "[Scattering forms and the positive geometry of kinematics, color, and the worldsheet](https://arxiv.org/abs/1711.09102)" by N. Arkani-Hamed, Y. Bai, S. He, and G. Yan


(2) "[Labelled tree graphs, Feynman diagrams and disk integrals](https://arxiv.org/abs/1708.08701)" by X. Gao , S. He , Y. Zhang


(3) "[Scattering from geometries](https://icts.ustc.edu.cn/chinese/seminar/transparencies/Song%20He/2018.5.25.html)" by S. He.


From (1):


"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."




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**Edit (5/29/2022)**:


I'd say my gut instinct in posing the original question--in spite of the close votes (since removed) and numerous downvotes--that the scattering amplitudes are deeply related to the combinatorics of associahedra, noncrossing partitions, and related OEIS arrays are vindicated by this month's publication of "[Connecting Scalar Amplitudes using The Positive Tropical Grassmannian](https://arxiv.org/pdf/2205.02722.pdf)" by Cachazo and Umbert in which my OEIS contributions [A134264](https://oeis.org/A134264) (related to Lagrange inversion. noncrossing partitions, the Fuss-Catalan number sequences, and a refinement of the Narayana triangle) and [A338135](https://oeis.org/A338135) (a special case of A134264) play critical roles.


As in my comment of Dec 11, 2019, to this question and [my answer](https://mathoverflow.net/questions/73711/the-concept-of-duality/348164#348164) to the MO-Q "The concept of duality" the KLT relation is mentioned--ref 9--in the C & U paper. The Banerjee et al. ref below is ref 9 in C & U. Refs 1, 3, 10, 22, 23 in C & U are mentioned in my comments to my MO-Q "[Guises of the Stasheff polytopes, associahedra for the Coxeter A\_n root system?](https://mathoverflow.net/questions/184803/guises-of-the-stasheff-polytopes-associahedra-for-the-coxeter-a-n-root-system)", reffed in A134264. Ref 10 in C & U is also given in this answer above. Ref 28 in C & U is in A338135. And, of course, I've also reffed numerous previous papers by Cachazo in a number of posts.


