72 research outputs found
Polygon dissections and some generalizations of cluster complexes
Let be a Weyl group corresponding to the root system or .
We define a simplicial complex in terms of polygon dissections
for such a group and any positive integer . For , is
isomorphic to the cluster complex corresponding to , defined in \cite{FZ}.
We enumerate the faces of and show that the entries of its
-vector are given by the generalized Narayana numbers , defined
in \cite{Atha3}. We also prove that for any the complex is shellable and hence Cohen-Macaulay.Comment: 9 pages, 3 figures, the type D case has been removed, some
corrections on the proof of Theorem 3.1 have been made. To appear in JCT
Geometric realizations of the accordion complex of a dissection
Consider points on the unit circle and a reference dissection
of the convex hull of the odd points. The accordion complex
of is the simplicial complex of non-crossing subsets of the
diagonals with even endpoints that cross a connected subset of diagonals of
. In particular, this complex is an associahedron when
is a triangulation and a Stokes complex when
is a quadrangulation. In this paper, we provide geometric
realizations (by polytopes and fans) of the accordion complex of any reference
dissection , generalizing known constructions arising from
cluster algebras.Comment: 25 pages, 10 figures; Version 3: minor correction
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Alexander Duality and Rational Associahedra
A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong,
Williams, and the author initiated the systematic study of {\em rational
Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics
(which is in turn a generalization of classical Catalan combinatorics). The
latter paper gave two possible models for a rational analog of the
associahedron which attach simplicial complexes to any pair of coprime positive
integers a < b. These complexes coincide up to the Fuss-Catalan level of
generality, but in general one may be a strict subcomplex of the other.
Verifying a conjecture of Armstrong, Williams, and the author, we prove that
these complexes agree up to homotopy and, in fact, that one complex collapses
onto the other. This reconciles the two competing models for rational
associahedra. As a corollary, we get that the involution (a < b)
\longleftrightarrow (b-a < b) on pairs of coprime positive integers manifests
itself topologically as Alexander duality of rational associahedra. This
collapsing and Alexander duality are new features of rational Catalan
combinatorics which are invisible at the Fuss-Catalan level of generality.Comment: 23 page
Dissections, Hom-complexes and the Cayley trick
We show that certain canonical realizations of the complexes Hom(G,H) and
Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in
fact instances of the polyhedral Cayley trick. For G a complete graph, we then
characterize when a canonical projection of these complexes is itself again a
complex, and exhibit several well-known objects that arise as cells or
subcomplexes of such projected Hom-complexes: the dissections of a convex
polygon into k-gons, Postnikov's generalized permutohedra, staircase
triangulations, the complex dual to the lower faces of a cyclic polytope, and
the graph of weak compositions of an integer into a fixed number of summands.Comment: 23 pages, 5 figures; improved exposition; accepted for publication in
JCT
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