Eleni Tzanaki

Polygon dissections and some generalizations of cluster complexes

Let W be a Weyl group corresponding to the root system An-1 or Bn. We define a simplicial complex ΔWm in terms of polygon dissections for such a group and any positive integer m. For m= 1, ΔWm is isomorphic to the cluster complex corresponding to W, defined in [S. Fomin, A.V. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003) 977-1018]. We enumerate the faces of ΔWm and show that the entries of its h-vector are given by the generalized Narayana numbers NWm(i), defined in [C.A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005) 179-196]. We also prove that for any m ≥ 1 the complex ΔWm is shellable and hence Cohen-Macaulay.

The maximum number of faces of the minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of k-faces, 0≤k≤d-1, of the Minkowski sum, P₁+P₂+P₃, of three d-dimensional convex polytopes P₁, P₂ and P₃ in R^d, as a function of the number of vertices of the polytopes, for any d≥2. Expressing the Minkowski sum as a section of the Cayley polytope C of its summands, counting the k-faces of P₁+P₂+P₃ reduces to counting the (k+2)-faces of C which meet the vertex sets of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of r d-polytopes in R^d, where r≥d. For d≥4, the maximum values are attained when P₁, P₂ and P₃ are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.

The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of

The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes

k

Bijections between generalized Catalan families of types A and C

-faces,

On the multiplicative order of the roots of bXqr+1−aXqr+dX−c

0\le{}k\le{}d-1

A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes

, of the Minkowski sum,

On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

P_1+P_2+P_3

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

, of three

The maximum number of faces of the Minkowski sum of two convex polytopes

d