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Slicing cones in various ways with a plane generates conic sections identified geometrically as hyperbolas, parabolas, or ellipses and algebraically, when suitably rotated, as certain rescaled quadratic polynomials, or forms, in two variables.

The refined face partition polynomials of the permutohedra (OEIS A049019) and associahedra (or Stasheff polytopes, A133437) and partition polynomials for noncrossing partitions (A134264) can all be algebraically related by re-scaling the associated monomials per the calculus of compositional and multiplicative inversion of power series, so this represents the algebraic analogue of the rescaling of the quadratic equations for conics.

Which geometric structures, such as possibly the noncrossing hypertrees of McCammond, brick polytopes, or simply polygons, when "sliced" in different ways, provide  representations of permutohedra, associahedra, and noncrossing partitions?

Ancillary question: Is there a geometric complex (possibly the Whitehouse simplicial complex, related to phylogenetic trees, or the trees themselves) that can be associated to Lagrange (compositional) inversion of exponential generating functions (i.e., formal Taylor, or divided powers, series) and can be incorporated in this geometric scheme if one exists?

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2-truncated cubes provide the "slicings" to generate the associahedra or permutahedra.

See "Upper and lower bound theorems for graph-associahedra" by Buchstaber and Volodin, "Cubical realizations of flag nestohedra and Gal's conjecture" by Volodin, and "Geometric realization of $\gamma$-vectors of 2-truncated cubes" by Volodin.

There is a video of a cube being truncated to form a 3-D permutahedron (truncated octahedron) at a website maintained by Vera Viana.

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  • $\begingroup$ And "The diagonal of the associahedra" by Naruki Masuda, Hugh Thomas, Andy Tonks, Bruno Vallette arxiv.org/abs/1902.08059 $\endgroup$
    – Tom Copeland
    Commented Sep 7, 2019 at 13:41
  • $\begingroup$ See also "Diagonals on the Permutahedra, Multiplihedra and Associahedra" by Samson Saneblidze, Ronald Umble arxiv.org/abs/math/0209109 $\endgroup$
    – Tom Copeland
    Commented Sep 7, 2019 at 17:03
  • $\begingroup$ "Direct families of polytopes with nontrivial Massey products" by Victor Buchstaber and Ivan Limonchenko arxiv.org/abs/1811.02221 $\endgroup$
    – Tom Copeland
    Commented Mar 21, 2020 at 18:08
  • $\begingroup$ See also p. 16 of "Toric Topology of Stasheff Polytopes" by Buchstaber $\endgroup$
    – Tom Copeland
    Commented Sep 30, 2020 at 20:44
  • $\begingroup$ See Fig. 1.4 on p. 18 of "From permutahedra to associahedra, a walk through geometric and algebraic combinatorics" by Vincent Pilaud lix.polytechnique.fr/~pilaud/documents/reports/… $\endgroup$
    – Tom Copeland
    Commented Oct 25, 2020 at 20:24

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