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# Associahedron

The associahedron is the -dimensional generalization of the pentagon. It was discovered by Stasheff in 1963 and it is also known as the Stasheff polytope. The number of nodes in the -associahedron is equivalent to the number of binary trees with nodes, which is the Catalan number .

The associahedron is the basic tool in the study of homotopy associative Hopf spaces.

Loday (2004) provides the following method for associahedron construction. Take , the set of planar binary trees with leaves. Define as the number of leaves to the left of the th vertex and as the number of leaves to the right of the th vertex. For in , define

The -associahedron is then defined as the convex hull of .

The associahedron can be obtained by removing facets from the permutohedron, and is related to the cyclohedron and permutohedron.

Cyclohedron, Pentagon, Permutohedron, Polytope

This entry contributed by Bryan Jacobs

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## References

Baez, J. "The Associahedron and Little -Cubes Operads." http://math.ucr.edu/home/baez/hda/associahedron.html.Hemmi, Y. "Associahedron, Permutohedron." http://www.math.kochi-u.ac.jp/hemmi/kenkyuu/polytope.html.Hohlweg, C. and Lange, C. "Realizations of the Associahedron and Cyclohedron." 2 Dec 2005. http://arxiv.org/abs/math.CO/0510614.Loday, J.-L. "Realization of the Stasheff Polytope." Arch. Math. 83, 267-278, 2004.Markl, M. "Simplex, Associahedron, and Cyclohedron." 9 Jul 1997. http://arxiv.org/alg-geom/9707009/.Postnikov, A. "Permutohedra, Associahedra, and Beyond." http://www-math.mit.edu/~apost/papers/permutohedron.pdf.Starck, M. "3D Representations." http://www.ac-noumea.nc/maths/amc/polyhedr/3D-img_.htm.

Associahedron

## Cite this as:

Jacobs, Bryan. "Associahedron." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Associahedron.html