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.