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Abstract: In this paper we report on results of our investigation into the algebraicstructure supported by the combinatorial geometry of the cyclohedron. Our newgraded algebra structures lie between two well known Hopf algebras: theMalvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra ofbinary trees. Connecting algebra maps arise from a new generalization of theTonks projection from the permutohedron to the associahedron, which we discovervia the viewpoint of the graph associahedra of Carr and Devadoss. At the sametime that viewpoint allows exciting geometrical insights into themultiplicative structure of the algebras involved. Extending the Tonksprojection also reveals a new graded algebra structure on the simplices.Finally this latter is extended to a new graded Hopf algebra one-sided withbasis all the faces of the simplices.

Author: Stefan Forcey, Derriell Springfield


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