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We exhibit a canonical connection between maximal (0, 1)-fillings of a moon polyomino avoiding northeast chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity… (More)

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized… (More)

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized… (More)

Contents 0. Preface 1 1. Introduction 3 2. What is a cluster algebra? 6 3. Using quivers as cluster algebra seeds 14 4. Finite type and finite mutation type classifications 20 4.1. Finite mutation type classification 24 4.2. Skew-symmetrizable cluster algebra seeds of finite mutation type 25 4.3. Class sizes of finite and affine quiver mutation types 29… (More)

We exhibit a canonical connection between maximal (0, 1)-fillings of a moon polyomino avoiding northeast chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity… (More)

We study bijections { Set partitions of type X } ˜ −→ { Set partitions of type X } • either the number of crossings and of nestings, • or the cardinalities of a maximal crossing and of a maximal nesting. In type D, the results are obtained only in the case of non-crossing and non-nesting set partitions. In all types, we show in particular that non-crossing… (More)

We describe edge labelings of the increasing flip graph of a sub-word complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these… (More)

- CHRISTIAN STUMP
- 2008

First, we investigate a generalization of the area statistic on Dyck paths for all crystallographic reflection groups. In particular, we explore Dyck paths of type B together with an area statistic and a major index. Then, we construct bijections between non-nesting and reverse non-crossing partitions for types A and B. These bijections simultaneously send… (More)

We show that the vertex barycenter of generalized associahedra and permutahedra coincide for any finite Coxeter system.