113 research outputs found
Generic rectangulations
A rectangulation is a tiling of a rectangle by a finite number of rectangles.
The rectangulation is called generic if no four of its rectangles share a
single corner. We initiate the enumeration of generic rectangulations up to
combinatorial equivalence by establishing an explicit bijection between generic
rectangulations and a set of permutations defined by a pattern-avoidance
condition analogous to the definition of the twisted Baxter permutations.Comment: Final version to appear in Eur. J. Combinatorics. Since v2, I became
aware of literature on generic rectangulations under the name rectangular
drawings. There are results on asymptotic enumeration and computations
counting generic rectangulations with n rectangles for many n. This result
answers an open question posed in the rectangular drawings literature. See
"Note added in proof.
Universal geometric cluster algebras
We consider, for each exchange matrix B, a category of geometric cluster
algebras over B and coefficient specializations between the cluster algebras.
The category also depends on an underlying ring R, usually the integers,
rationals, or reals. We broaden the definition of geometric cluster algebras
slightly over the usual definition and adjust the definition of coefficient
specializations accordingly. If the broader category admits a universal object,
the universal object is called the cluster algebra over B with universal
geometric coefficients, or the universal geometric cluster algebra over B.
Constructing universal coefficients is equivalent to finding an R-basis for B
(a "mutation-linear" analog of the usual linear-algebraic notion of a basis).
Polyhedral geometry plays a key role, through the mutation fan F_B, which we
suspect to be an important object beyond its role in constructing universal
geometric coefficients. We make the connection between F_B and g-vectors. We
construct universal geometric coefficients in rank 2 and in finite type and
discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure
A combinatorial approach to scattering diagrams
Scattering diagrams arose in the context of mirror symmetry, but a special
class of scattering diagrams (the cluster scattering diagrams) were recently
developed to prove key structural results on cluster algebras. We use the
connection to cluster algebras to calculate the function attached to the
limiting wall of a rank-2 cluster scattering diagram of affine type. In the
skew-symmetric rank-2 affine case, this recovers a formula due to Reineke. In
the same case, we show that the generating function for signed Narayana numbers
appears in a role analogous to a cluster variable. In acyclic finite type, we
construct cluster scattering diagrams of acyclic finite type from Cambrian fans
and sortable elements, with a simple direct proof.Comment: This is the second half of arXiv:1712.06968, which was originally
titled "Scattering diagrams and scattering fans". The contents of this paper
will be removed from arXiv:1712.06968, which will be re-titled "Scattering
fans." Version 2: Minor expository changes. (We thank some anonymous referees
for helpful comments.
Cambrian Lattices
For an arbitrary finite Coxeter group W we define the family of Cambrian
lattices for W as quotients of the weak order on W with respect to certain
lattice congruences. We associate to each Cambrian lattice a complete fan,
which we conjecture is the normal fan of a polytope combinatorially isomorphic
to the generalized associahedron for W. In types A and B we obtain, by means of
a fiber-polytope construction, combinatorial realizations of the Cambrian
lattices in terms of triangulations and in terms of permutations. Using this
combinatorial information, we prove in types A and B that the Cambrian fans are
combinatorially isomorphic to the normal fans of the generalized associahedra
and that one of the Cambrian fans is linearly isomorphic to Fomin and
Zelevinsky's construction of the normal fan as a "cluster fan." Our
construction does not require a crystallographic Coxeter group and therefore
suggests a definition, at least on the level of cellular spheres, of a
generalized associahedron for any finite Coxeter group. The Tamari lattice is
one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are
identified and characterized in terms of signed pattern avoidance. We also show
that open intervals in Cambrian lattices are either contractible or homotopy
equivalent to spheres.Comment: Revisions in exposition (partly in response to the suggestions of an
anonymous referee) including many new figures. Also, Conjecture 1.4 and
Theorem 1.5 are replaced by slightly more detailed statements. To appear in
Adv. Math. 37 pages, 8 figure
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