600 research outputs found
The Catalan simplicial set
The Catalan numbers are well-known to be the answer to many different
counting problems, and so there are many different families of sets whose
cardinalities are the Catalan numbers. We show how such a family can be given
the structure of a simplicial set. We show how the low-dimensional parts of
this simplicial set classify, in a precise sense, the structures of monoid and
of monoidal category. This involves aspects of combinatorics, algebraic
topology, quantum groups, logic, and category theory.Comment: 15 pages. Replaces and expands upon parts of arXiv:1307.0265;
remaining parts of arXiv:1307.0265 will be incorporated into a sequel.
Version 2: minor revision; to appear in Math. Proc. Camb. Phil. So
Rational associahedra and noncrossing partitions
Each positive rational number x>0 can be written uniquely as x=a/(b-a) for
coprime positive integers 0<a<b. We will identify x with the pair (a,b). In
this paper we define for each positive rational x>0 a simplicial complex
\Ass(x)=\Ass(a,b) called the {\sf rational associahedron}. It is a pure
simplicial complex of dimension a-2, and its maximal faces are counted by the
{\sf rational Catalan number} \Cat(x)=\Cat(a,b):=\frac{(a+b-1)!}{a!\,b!}. The
cases (a,b)=(n,n+1) and (a,b)=(n,kn+1) recover the classical associahedron and
its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and
Fomin-Reading. We prove that \Ass(a,b) is shellable and give nice product
formulas for its h-vector (the {\sf rational Narayana numbers}) and f-vector
(the {\sf rational Kirkman numbers}). We define \Ass(a,b) via {\sf rational
Dyck paths}: lattice paths from (0,0) to (b,a) staying above the line y =
\frac{a}{b}x. We also use rational Dyck paths to define a rational
generalization of noncrossing perfect matchings of [2n]. In the case (a,b) =
(n, mn+1), our construction produces the noncrossing partitions of [(m+1)n] in
which each block has size m+1.Comment: 21 pages, 8 figure
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
A new graph invariant arises in toric topology
In this paper, we introduce new combinatorial invariants of any finite simple
graph, which arise in toric topology. We compute the -th (rational) Betti
number and Euler characteristic of the real toric variety associated to a graph
associahedron P_{\B(G)}. They can be calculated by a purely combinatorial
method (in terms of graphs) and are named and , respectively. To
our surprise, for specific families of the graph , our invariants are deeply
related to well-known combinatorial sequences such as the Catalan numbers and
Euler zigzag numbers.Comment: 21 pages, 3 figures, 4 table
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
Stokes posets and serpent nests
We study two different objects attached to an arbitrary quadrangulation of a
regular polygon. The first one is a poset, closely related to the Stokes
polytopes introduced by Baryshnikov. The second one is a set of some paths
configurations inside the quadrangulation, satisfying some specific
constraints. These objects provide a generalisation of the existing
combinatorics of cluster algebras and nonnesting partitions of type A.Comment: 24 pages, 12 figure
- …