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Minkowski decomposition of associahedra and related combinatorics. (English) Zbl 1283.52014
Summary: 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 F. Ardila et al. [Discrete Comput. Geom. 43, No. 4, 841–854 (2010; Zbl 1204.52016)]. 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.

MSC:
 52B10 Three-dimensional polytopes
polymake
Full Text:
References:
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