Let $W$ be a Weyl group corresponding to the root system $A_{n-1}$ or $B_n$.
We define a simplicial complex $ Δ^m_W $ in terms of polygon dissections for such a group and any positive integer $m$. For $ m=1 $, $ Δ^m_W$ is isomorphic to the cluster complex corresponding to $ W $, defined in \cite{FZ}. We enumerate the faces of $ Δ^m_W $ and show that the entries of its $h$-vector are given by the generalized Narayana numbers $ N^m_W(i) $, defined in \cite{Atha3}. We also prove that for any $ m \geq 1$ the complex $ Δ^m_W $ is shellable and hence Cohen-Macaulay.
This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some known enumerations are recovered and several new ones are provided.
We formulate the following general problem. To the best of our knowledge, it is open and has not previously been considered. It seems (at least to us!) to be difficult; at any rate, more difficult than enumerating the dissections themselves. Enumerate the distinct quiddities of dissections of the $N$-gon into $m$ cells.
The Catalan numbers $C_k$ were first studied by Euler, in the context of enumerating triangulations of polygons $P_{k+2}$. Among the many generalizations of this sequence, the Fuss-Catalan numbers $C^{(d)}_k$ count enumerations of dissections of polygons $P_{k(d-1)+2}$ into $(d+1)$-gons. In this paper, we provide a formula enumerating polygonal dissections of $(n+2)$-gons, classified by partitions $λ$ of $[n]$. We connect these counts $a_λ$ to reverse series arising from iterated polynomials. Generalizing this further, we show that the coefficients of the reverse series of polynomials $x=z-\su…
We give a short proof for a formula for the number of divisions of a convex (sn+2)-gon along non-crossing diagonals into (sj+2)-gons, where 1<=j<=n-1. In other words, we consider dissections of an (sn+2)-gon into pieces which can be further subdivided into (s+2)-gons. This formula generalizes the formulas for classical numbers of polygon dissections: Euler-Catalan number, Fuss number and Kirkman-Cayley number. Our proof is elementary and does not use the method of generating functions.
Let $Δ=\{ δ_1,δ_2,...,δ_m \} $ be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function $D_{Δ}(z,u_1,u_2,...,u_m),$ which counts polygon dissections and marks subgraph copies of $δ_i$ with the variable $u_i$. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on $Δ$. The decomposition also gives a defining system for $D_{Δ}(z,\mathbf{0})$, which encodes polygon dissections that restrict these patterns as subgraphs. In this way, we are able to extract normal limit laws for the patterns when the…
We study a $2 \times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $\mathrm{PSL}(2,\mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,\mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,\mathbb{Z})$.
The logical depth of a graph $G$ is the minimum quantifier depth of a first order sentence defining $G$ up to isomorphism in the language of the adjacency and the equality relations. We consider the case that $G$ is a dissection of a convex polygon or, equivalently, a biconnected outerplanar graph. We bound the logical depth of a such $G$ from above by a function of combinatorial parameters of the dual tree of $G$.
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, a…
We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array} \right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots \left( \begin{array}{cc} a_1&-1 1&0 \end{array} \right) $ is either the identity matrix $\mathrm Id$, its negative $-\mathrm Id$, or square root of $-\mathrm Id$. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.
For integers m, n $\ge$ 1, we describe a bijection sending dissections of the (mn + 2)-regular polygon into (m + 2)-sided polygons to a new basis of the quotient of the polynomial algebra in mn variables by an ideal generated by some kind of higher quasi-symmetric functions. We show that divisibility of the basis elements corresponds to a new partial order on dissections, which is studied in some detail.
Consider 2n points on the unit circle and a reference dissection D of the convex hull of the odd points. The accordion complex of D is the simplicial complex of subsets of pairwise noncrossing diagonals with even endpoints that cross a connected set of diagonals of the dissection D. In particular, this complex is an associahedron when D is a triangulation, and a Stokes complex when D is a quadrangulation. We exhibit a bijection between the facets of the accordion complex of D and some dual objects called the serpent nests of D, settling in particular a conjecture stated by F.~Chapoton (2016) i…
The matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jelić Milutinović et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.
Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.
The central component of a polygon triangulation is defined as the triangle or diameter that contain its geometric center. More generally, every polygon dissection contains a central component. Using this notion, we derive new recurrences for the Catalan and $k$-Catalan numbers, and use these recursions to prove congruence relations of these numbers. We also enumerate the triangulations that contain a fixed vertex in their central components.
The paper establishes that the rank of a regular polygonal complex in 3-space E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in 3-space are the eight regular 4-apeirotopes.
Coxeter defined the notion of frieze pattern, and Conway and Coxeter proved that triangulations of polygons are in bijection with integral frieze patterns. We show a $p$-angulated generalisation involving non-integral frieze patterns. We also show that polygon dissections give rise to even more general non-integral frieze patterns.
The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the classification of regular polygonal complexes, chiral polyhedra, and more generally, two-orbit polyhedra.
Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most…
Catacondensed benzenoids (those benzenoids having no carbon atom belonging to three hexagonal rings) form the simplest class of polycyclic aromatic hydrocarbons (PAH). They have a long history of study and are of wide chemical importance. In this paper, mathematical possibilities for natural extension of the notion of a catacondensed benzenoid are discussed, leading under plausible chemically and physically motivated restrictions to the notion of a catacondensed chemical hexagonal complex (CCHC). A general polygonal complex is a topological structure composed of polygons that are glued togethe…
Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realiz…
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated generating function, of the multipath complex for some families of graphs, including transitive tournaments and complete bipartite graphs. Then, we compute the homotopy type of multipath complexes of linear graphs, polygons, small grids and transitive tournaments. We show that they are all contractible or wedges of spheres. We introduce a new technique for deco…
We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface. We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, differ…
A~$k$-associahedron is a simplicial complex whose facets, called~$k$-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no~$k+1$ diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This paper reports on conjectural realizations for all~$2$-associahedra, obtained by heur…
Using a result of Vdovina, we may associate to each complete connected bipartite graph $κ$ a $2$-dimensional square complex, which we call a tile complex, whose link at each vertex is $κ$. We regard the tile complex in two different ways, each having a different structure as a $2$-rank graph. To each $2$-rank graph is associated a universal C*-algebra, for which we compute the K-theory, thus providing a new infinite collection of $2$-rank graph algebras with explicit K-groups. We determine the homology of the tile complexes, and give generalisations of the procedures to complexes and systems c…
