An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

(This is a duplicate of a Mathoverflow question posed in Oct 2014 that ran the gauntlet of the OCD cadre there–the demonstrative ones I assume avoid stepping on cracks in the sidewalk and become obstructive, hostile, and/or jealous at popular questions they can’t immediately circumscribe within their own rigid, limited views on how and what math should be done. See also this blog post and this MO-Q. In a couple of days the question got around 800 views, 9 upvotes, 7 downvotes, two or three close requests, several unconstructive/petty comments on rewriting/formatting, but no answers. Recognizing, from experience on MO, the futility of further refining the question and having pursued a plausible partial answer myself, I deleted most of it. Here is the full text so that I and others can easily follow related references.)

Prelude on Two Threads

Last month (Sept. 2014) the workshop New Geometric Structures in Scattering Amplitudes was hosted by CMI with the following partial overview:

Recently, remarkable mathematical structures have emerged in the study of scattering amplitudes, revealing deep links to algebraic geometry, arithmetic and combinatorics. There have been many exciting dual representations of amplitudes including

  •  polytopes, twistor diagrams and the positive grassmannian leading to the concept of the amplituhedron,
  •  polylogs, multizeta values and multiloop integrals,
  •  …

Each of these disparate ideas and methods have their own remarkable successes, and indeed have led to important progress in related areas of mathematics. They also face obstacles that they need to overcome in order to become important tools in resolving the most important open conjectures in the field and for the explicit construction of amplitudes.

Classic Number Clans in the Tapestry

It’s a rich tapestry in the making (note this Dec. workshop), weaving together much interesting math and physics, in which I would like to follow some threads particularly intriguing to me (since they relate to different perspectives on the combinatorics of polytopes and simplicial complexes related to Lagrange compositional inversion).

The two bullets above relate to two threads

I ) T_V: Totally non-negative Grassmannians G^+(k,n) ~ matroid polytopes P_M ~ Vol(P_M) ~ degrees of toric varieties ~ number of solutions of polynomial equations related to scattering amplitudes in twistor string theory from volume/contour integrals over G^+

II) T_C: G^+ ~ Stasheff associahedra ~ cluster algebras/coordinates ~ generalized polylogarithms ~ MZVs ~ multiloop integrals for scattering amplitudes.

It was Marni Sheppeard through her paper “Constructive Motives and Scattering]” who first gave me a docent’s tour of this tapestry. In particular, she points out some threads interweaving Grassmannians, associahedra, cluster algebra, generalized permutohedra, volumes of hypersimplices and the Eulerian numbers, volumes (and binary trees) enumerated by the Narayana numbers, and scattering amplitudes, among others.

And, Lauren Williams in “Enumeration of totally positive Grassmann cells]” develops a polynomial generating function A_{k,n}(q) whose q^d coefficient is the number of totally positive cells in G^+(k,n) that have dimension d and goes on to show that for the binomial transform \hat{E}_{k,n}(q)=q^{k-n}\sum^n_{i=0} (-1)^i \binom{n}{i} A_{k,n-i}(q) that \hat{E}_{k,n(}(1)=E_{k,n}, the Eulerians, and \hat{E}_{k,n}(0)=N_{k,n}, the Narayanaians. She reiterates this in her presentation “The Positive Grassmannian (a mathematician’s perspective)” and relates G+ to soliton shallow-water-wave solutions of a KP equation, noting the roles of G^+ in computing scattering amplitudes in string theory, a relation to free probability, and the occurrence of the Eulerians and Narayanaians in the BCFW recurrence and twistor string theory.

The number clans that appear in the tapestry are listed below along with some associations. (The Wardians seem peripheral for the moment, but they do lead to the associahedra through fans and phylogenetic trees, and the refined ones can be scaled to the refined f-vectors of the associahedra through their relation to Lagrange inversion.)

Some relations to number clans:

Eulerians, E_{n,k} (A008292, refined-A145271):

h-vectors of the permutohedra ~ volumes of hypersimplices ~ degrees of varieties ~ number of solutions of polynomials for scattering amplitudes

Catalanians, C_{n,k} (A033282, refined-A133437):

f-vectors of Stasheff associahedra (for Coxeter group A_n), related to dissections of polygons (with the Catalans, # of vertices, enumerating the triangulations) ~ structure of associahedra reflects cluster algebra relations

Narayanaians, N_{n,k} (A001263, refined-A134264):

h-vectors of the simplicial complex dual to the Stasheff associahedra, sum to the Catalan numbers, enumerate non-crossing partitions on [n] and refinement of binary trees (right-pointing leaves), refined Narayanaians relate number of connected positroids on [n] to the total number of positroids through an inversion (see A134264)

Wardians, W{n,k} (A134991, refined-A134685):

f-vectors of the Whitehouse simplicial complex associated with the tropical Grassmannians G(2,k) and phylogenetic trees (Bergman matroids?), related to enumeration of partitions of 2n objects.


A) Are there other perspectives on this tapestry involving these classic number clans, i.e., other ways in which these clans show up in the tapestry?

B) Can someone give a more cogent overview of these two threads and their relation to the classic number arrays?


(More details for the interested.)

Thread T_V:

1) Alcoved polytopes I, Lam and Postnikov, pages 1, 2, 18, and 21,

keywords: volumes, hypersimplices, Eulerian, grassmannian manifold, torus orbit

2) Matroid polytopes and volumes, Ardila, Benedetti, and Doker, page 6,

keywords: generalized permutohedra, grassmannian, torus orbit, volumes

3) Loops, Legs and Twistors, Spradlin,

keywords: contour integral, amplitudes, polynomials equations, Eulerian numbers

4) A note on polytopes for scattering amplitudes, Arkani-Hamed, Bourjaily, Cachazo, Hodges, and Trnka, pages 5-10,

keywords: twistor theory, volumes, areas, contour integral, differential forms

5) Scattering in three dimensions from rational maps, Cachazo, He, and Yuan, pages 4 and 11,

keywords: Eulerian numbers, scattering equations

6) Scattering Equations, Yuan

keywords: vanishing, quadratic differential, polynomial maps, Eulerian numbers

7) Gravity in Twistor Space and its Grassmannian Formulation, Cachazo, Mason, and Skinner, pages 18 and 19,

keywords: Eulerian number, marked points

Thread T_C;

8) Matching polytopes, toric geometry, and the non-negative part of the Grassmannian, Postnikov, Speyer, and Williams, pages 1-2 and 12-13,

keywords: Grassmannians, matroid polytope, toric variety, cluster algebra

9) Cluster Polylogarithms for Scattering Amplitudes, Golden, Paulos, Spradlin, and Volovich, pages 8-13,

keywords: Stasheff associahedra, cluster functions

10) Studying Quantum Field Theory, Todorov, pages 17-20,

keywords: Catalan numbers, iterated integrals, simplices, polylogarithms

Additional notes on number clans, combinatorics, polytopes, and algebraic geometry:

11) On the Hilbert polynomials and Hilbert series of homogeneous projective varieties, Gross and Wallach, pages 13 and 14,

keywords: Grassmannian, Catalan numbers, Narayana numbers

12) Eulerian polynomials, Hirzebruch

13) New moduli spaces of pointed curves and pencils of flat connections, Losev and Manin, page 8, (Eulerians misnamed as the the Euler numbers)

14) For the Eulerian polynomial, see my recent entry in the formula section of A008292 and the associated links to Lenart and Zanoulline, and Buchstaber and Bunkova.

15) For the Narayana polynomials, see my recent example in A134264 and the associated reference to Ardila, Rincon, and Williams.

16) For enumeration of positroid cells of G+ and generating series interpolating between the Eulerians and Narayanaians, see A046802 and links therein.

17) Reflection group counting and q-counting, Reiner,

keywords: Catalan and Narayana numbers, parking functions, Weyl groups, q-extensions

Examining these brings to light another colorful thread in the tapestry related to moduli spaces, configuration spaces of particles, marked surfaces, and the polytopes associated to Lagrange inversion in different “coordinates”–o.g.f.s, e.g.f.s, etc. (The Lagrange inversion associated with the refined Eulerian partition polynomials seems to be “coordinate-free” in terms of the input in some sense.) Rather intriguing to me.

Refs added Nov. 2016:

18) Hedgehog Bases for A_n Cluster Polylogarithms  … , Parker, Scherlis, Spradlin, Volovich

19) Totally nonnegative Grassmannian and Grassmann polytopes, Lam

20) OEIS A248727: face-vectors of stellahedra / stellohedra, whose h-vectors enumerate positroid cells of the totally nonnegative Grassmannian. Related to the Eulerians.


22) A131758 gives relations among the Eulerian numbers and polylogarithms of negative orders (see Wikipedia also).

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11 Responses to An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

  1. Tom Copeland says:

    From 4gravitons ( remarking on talks at the Amplitudes 2017 conference: Between the two of them, Nima and Yuntao 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.

  2. Tom Copeland says:

    See the modified Mathoverflow question for more references: (Now 10 upvotes and seven downvotes.)

  3. Tom Copeland says:

    On the Eulerians, see “Scattering Equations: Real Solutions and Particles on a Line” by Freddy Cachazo, Sebastian Mizera, Guojun Zhang.

  4. Tom Copeland says:

    Other early refs: “Motivic Amplitudes and Cluster Coordinates” by John Golden, Alexander B. Goncharov, Marcus Spradlin, Cristian Vergu, Anastasia Volovich and Another recent talk by He is “Scattering Forms from Geometries at Infinity”

  5. Tom Copeland says:

    And I thought I had problems with certain gatekeepers. Read the revisionist historical account “Grothendieck: The Myth of a Break” by Lobry. As in any community, the math community encompasses your full spectrum of human personalities and group dynamics–from intolerant, obstructionist cliques to tolerant, nurturing facilitators.

  6. Tom Copeland says:

    See “Biadjoint scalar tree amplitudes and intersecting dual associahedra” by Hadleigh Frost (, “Combinatorics and Topology of Kawai-Lewellen-Tye Relations” by Sebastian Mizera (, and

  7. Tom Copeland says:

    Currently the associated MO-Q has 11 upvotes and eight downvotes while a minor cottage industry has developed in the intervening years since the posting producing many papers on the relations between associahedra and scattering amplitudes in several field theories. I even notified Dirk Kreimer and Karen Yeats via email in Nov 2016 of the appearance of the refined face polynomials of the associahedra ( in their 2016 paper “Diffeomorphisms of Quantum Fields” of which they responded they had been unaware. Karen Yeats through later joint publications and discussions with colleagues contributed to development of relations among the combinatorics of associahedra and scattering amplitudes.

  8. Tom Copeland says:

    Now 12 upvotes and 8 downvotes. Would love to know the profiles of the two sets of voters. Bet there is a lot to be gleaned from the statistics of the users’ questions (views, likes, favorites, and other data), answers, and voting on the signatures/indicators of obstructionists and facilitators.

  9. Tom Copeland says:

    “Positive configuration space” by
    Nima Arkani-Hamed, Thomas Lam, Marcus Spradlin

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