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Relating the associahedron and the permutohedron. (English) Zbl 0873.51016
Loday, Jean-Louis (ed.) et al., Operads: Proceedings of renaissance conferences. Special session and international conference on moduli spaces, operads, and representation theory/operads and homotopy algebra, March 1995/May–June 1995, Hartford, CT, USA/Luminy, France. Providence, RI: American Mathematical Society. Contemp. Math. 202, 33-36 (1997).
It was shown by M. M. Kapranov [J. Pure Appl. Algebra 85, No. 2, 119-142 (1993; Zbl 0812.18003)] that the combinatorics of the permutohedra and associahedra can be combined to give a family of polytopes, the permutoassociahedra. In this short note a slightly different point of view is emphasized: the associahedra can themselves be seen as retracts of the permutohedra. A natural cellular quotient map from the permutohedron $${\mathcal P}_n$$ to the associahedron $${\mathcal K}_{n+1}$$ is constructed.
For the entire collection see [Zbl 0855.00018].

##### MSC:
 51M20 Polyhedra and polytopes; regular figures, division of spaces
##### Keywords:
permutations; combinatorics; posets; permutohedra; associahedra