Return to Question

Edit (Dec 2017): In a comment below Ian Agol references a recent paper by Aguiar and Ardila in which the following assertion can be found applying to generalized permutahedra of which Loday's realization of the Stasheff associahedra is one example:

Edit (Dec 2017): In a comment below Ian Agol references the recent paper "Hopf monoids and generalized permutahedra" by Aguiar and Ardila, in which the following assertion can be found applying to generalized permutahedra of which Loday's realization of the Stasheff associahedra is one example:

Edit (Dec 2017): In a comment below Ian Agol references a recent paper by Aguiar and Ardila in which the following assertion can be found applying to generalized permutahedra of which Loday's realization of the Stasheff associahedra is one example:

Generalized permutahedra arise in a multitude of settings, and can be used to model many combinatorial objects: graphs, matroids, posets, set partitions, paths, and many others. In this section we present one reason for the ubiquity of these polyhedra: generalized permutahedra are equivalent to submodular functions, which are central objects in optimization. These functions occur in numerous mathematical and real-world contexts, since they are characterized by a diminishing returns property that is natural in many settings.

This correspondence between the refined f-vectors of the n-Dim Stasheff polytope, or associahedron, and the coefficients of the (n+2)-th term of the compositional inverse holds in general, (see A133437, inversion for power series, and compare with A033282, coarse f-vectors for associahedra, and with MO-6373).

Second viewpoint: Stasheff associahedra are intimately related to the moduli spaces of colliding particles (Devadoss, Devadoss/Heath/Vipismakul, Devadoss/Fehrman/Heath/Vashist). String interactions generate the moduli spaces of Riemann surfaces (Zwiebach, A First Course in String Theory, pg. 310) with punctures corresponding to particles interacting on a line segment. There is much literature on the relations among compositional inversion/Legendre transformation, Feynman functional/path/gaussian integrals representing partition functions and sums over Feynman diagrams/graphs for point particle interactions (Connes/Marcolli's "Noncommutative Geometry, Quantum Fields and Motives" pg. 51, Borcherd pg. 34, Getzler, Manin, Abdesselam, Bergstrom and Brown). Are there analogous arguments directly in terms of sums over moduli spaces for string interactions [as for the beta integral for the Veneziano amplitudes (Zwiebach, pg. 311)] that circumvent the Feynman particle/stable graph interpretations and highlight more directly the connections between compositional inverses/Legendre transforms and the face polynomials of associahedra?
(See also MOQ 22291 and make the change of variables $x=f^{-1}(y)$ in Theo's integral and maybe a Wick rotation.)

This correspondence between the refined f-vectors of the n-Dim Stasheff polytope, or associahedron, and the coefficients of the (n+2)-th term of the compositional inverse holds in general, (see A133437, inversion for power series, and compare with A033282, coarse f-vectors for associahedra, and with MO-6373).

Second viewpoint: Stasheff associahedra are intimately related to the moduli spaces of colliding particles (Devadoss, Devadoss/Heath/Vipismakul, Devadoss/Fehrman/Heath/Vashist). String interactions generate the moduli spaces of Riemann surfaces (Zwiebach, A First Course in String Theory, pg. 310) with punctures corresponding to particles interacting on a line segment. There is much literature on the relations among compositional inversion/Legendre transformation, Feynman functional/path/gaussian integrals representing partition functions and sums over Feynman diagrams/graphs for point particle interactions (Connes/Marcolli's "Noncommutative Geometry, Quantum Fields and Motives" pg. 51, Borcherd pg. 34, Getzler, Manin, Abdesselam, Bergstrom and Brown). Are there analogous arguments directly in terms of sums over moduli spaces for string interactions [as for the beta integral for the Veneziano amplitudes (Zwiebach, pg. 311)] that circumvent the Feynman particle/stable graph interpretations and highlight more directly the connections between compositional inverses/Legendre transforms and the face polynomials of associahedra?
(See also MOQ 22291 and make the change of variables $x=f^{-1}(y)$ in Theo's integral and maybe a Wick rotation.)