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, Benedetti & Doker (2010). The coefficients yI of such a Minkowski decomposition can be computed by Möbius inversion if tight right-hand sides zI 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: • how to compute the tight value zI for any inequality that is redundant for an associahedron but facet-defining for the classical permutahedron. More precisely, each value zI is described in terms of tight values zJ of facet-defining inequalities of the corresponding associahedron determined by combinatorial properties of I. • the computation of the values yI of Ardila, Benedetti & Doker can be significantly simplified and depends on at most four values za(I), zb(I), zc(I) and zd(I). • 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. • a combinatorial interpretation of the values yI using a labeled n-gon. This result is inspired from similar interpretations for vertex coordinates originally described by J.-L. Loday and well-known interpretations for the zI -values of facet-defining inequalities.