Discrete & Computational Geometry

, Volume 50, Issue 4, pp 903–939 | Cite as

Minkowski Decomposition of Associahedra and Related Combinatorics



Realisations of associahedra with linear non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila et al. (Discret Comput Geom, 43:841–854, 2010). The coefficients \(y_I\) of such a Minkowski decomposition can be computed by Möbius inversion if tight right-hand sides \(z_I\) are known not just for the facet-defining inequalities of the associahedron but also for all inequalities of the permutahedron that are redundant for the associahedron. We show for certain families of these associahedra: (1) How to compute the tight value \(z_I\) for any inequality that is redundant for an associahedron but facet-defining for the classical permutahedron. More precisely, each value \(z_I\) is described in terms of tight values \(z_J\) of facet-defining inequalities of the corresponding associahedron determined by combinatorial properties of \(I\). (2) The computation of the values \(y_I\) of Ardila, Benedetti & Doker can be significantly simplified and depends on at most four values \(z_{a(I)},\,z_{b(I)},\,z_{c(I)}\) and \(z_{d(I)}\). (3) The four indices \(a(I),\,b(I),\,c(I)\) and \(d(I)\) are determined by the geometry of the normal fan of the associahedron and are described combinatorially. (4) A combinatorial interpretation of the values \(y_I\) using a labeled \(n\)-gon. This result is inspired from similar interpretations for vertex coordinates originally described by Loday and well-known interpretations for the \(z_I\)-values of facet-defining inequalities.


Realizations of Polytopes Associahedra Minkowski sums and differences Möbius inversion 


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.FB Mathematik & Informatik, Freie Universität BerlinBerlinGermany
  2. 2.Institut de Mathématiques de Jussieu, Université Paris VIParisFrance

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