Dissections Of Polygons Research Articles

Quiddities of polygon dissections and the Conway-Coxeter Frieze equation

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.

Counting Quiddities of Polygon Dissections

Counting Quiddities of Polygon Dissections

Friezes, weak friezes, and T-paths

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables.This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula.We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued.

Quelques éléments de combinatoire des matrices de [formula omitted

A Theorem of V. Ovsienko characterizes sequences of positive integers (a1,a2,…,an) such that the (2×2)-matrix (an−110)⋯(a1−110) is equal to ±Id. In this paper, we study this equation when we replace ±Id by ±M. In particular, we give a combinatorial description of the solutions of this equation in terms of dissections of convex polygons in the cases M=(0−110) and M=(1101).

Farey Boat: Continued Fractions and Triangulations, Modular Group and Polygon Dissections

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 $\mathrm{PSL}(2,\mathbb {Z})$ . 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 $\mathrm{PSL}(2,\mathbb {Z})$ .

Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs

Let Δ={δ1,δ2,…,δm} be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function DΔ(z,u1,u2,…,um), which counts polygon dissections and marks subgraph copies of δi with the variable ui. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on Δ. The decomposition also gives a defining system for DΔ(z,0), which encodes polygon dissections that avoid these patterns as subgraphs. In this way, we are able to extract normal limit laws for the patterns when they are encoded, and perform asymptotic enumeration of the resulting classes when they are avoided. The results can be transferred to the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.

Partitions of unity in $$\mathrm {SL}(2,\mathbb Z)$$ SL ( 2 , Z ) , negative continued fractions, and dissections of polygons

We characterize sequences of positive integers $$(a_1,a_2,\ldots ,a_n)$$ for which the $$2\times 2$$ matrix $$\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) $$ 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.

A p-angulated Generalisation of Conway and Coxeter’s Theorem on Frieze Patterns

Abstract 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 nonintegral frieze patterns. We also show that polygon dissections give rise to even more general nonintegral frieze patterns.

Scaling limits of random outerplanar maps with independent link-weights

La limite d’échelle des cartes planaires extérieures simples a été établie par Caraceni via une bijection de Bonichon, Gavoille et Hanusse. Dans ce papier, nous construisons une nouvelle bijection entre les cartes planaires extérieures, et les arbres décorés par des suites ordonnées de dissections de polygones enracinées. Nous utilisons cette décomposition pour obtenir une nouvelle preuve, plus courte, du résultat de Caraceni, qui s’étend en outre au cadre général de la percolation de premier passage. Nous obtenons des bornes précises sur la queue de distribution du diamètre des cartes planaires extérieures, et retrouvons les formules d’énumération asymptotiques de ces cartes. Nos méthodes nous permettent également de traiter le cas de sous-classes comme les cartes planaires extérieures biparties.

Polygonal dissections and reversions of series

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 $\lambda$ of $[n]$. We connect these counts $a_{\lambda}$ to reverse series arising from iterated polynomials. Generalizing this further, we show that the coefficients of the reverse series of polynomials $x=z-\sum_{j=0}^{\infty} b_j z^{j+1}$ enumerate colored polygonal dissections.

Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric $r$-dissections of polygons and planar $S$-dissections are obtained.

The CRT is the scaling limit of random dissections

Abstract–We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform p-angulations. As their number of vertices n goes to infinity, we show that these random graphs, rescaled by , converge in the Gromov–Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 304–327, 2015

Random stable laminations of the disk

We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous’ Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Lévy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.

On symmetric quadrangulations and triangulations

This article presents new enumerative results related to symmetric planar maps. In the first part a new way of enumerating rooted simple quadrangulations and rooted simple triangulations is presented, based on the description of two different quotient operations on symmetric simple quadrangulations and triangulations. In the second part, based on the results of Bouttier, Di Francesco and Guitter and on quotient and substitution operations, the series of three families of symmetric quadrangular and triangular dissections of polygons are computed, with control on the distance from the central vertex to the outer boundary.

AVOIDING PERMUTATIONS AND THE NARAYANA NUMBERS

We study 132 avoiding permutations that also avoid <TEX>$(2r+1)(2r+2){\cdots}12$</TEX> but contain <TEX>$(2r-1)(2r){\cdots}12$</TEX> pattern. We find an identity between the number of these permutations and the Narayana number. We also present relations between 132 avoiding permutations and polygon dissections. Finally, a generalization of these permutations is obtained.

Beyond Swinging: Hinged Dissections that Twist or Fold

A geometric dissection is a cutting of a geometric figure into pieces that can be rearranged to form another figure. Some dissections can be connected with hinges so that the pieces form one figure when swung one way on the hinges, and form the other figure when swung another way. In addition to using “swing hinges”, which allow rotation in the plane, we can use “twist hinges”, which allow one piece to be flipped over relative to another piece via rotation by 180° through a third dimension. Furthermore, we can use “fold hinges”, which allow rotation along a shared edge, a motion that is akin to folding. This talk will introduce a variety of twist-hinged and fold-hinged dissections of regular polygons and stars, and other figures such as polyominoes. The emphasis will be on both appreciating and understanding these fascinating mathematical recreations. I will employ algorithmic and tessellation-based techniques, as well as symmetry and other geometric properties, to design the dissections. The goal will be to minimize the number of pieces, subject to the dissection being suitably hinged. Animations and video will be used to demonstrate the hinged dissections, in addition to actual physical models. Bio: Greg N. Frederickson is a Professor of Computer Science at Purdue University, in West Lafayette, Indiana. His primary area of research is the design and analysis of algorithms, and he has served on the editorial boards of SIAM Journal on Computing, SIAM Journal on Discrete Mathematics, Algorithmica, and IEEE Transactions on Computers. He also pursues interests in mathematical recreations, specifically geometric dissection. On this topic he has published three books and a number of articles. He has twice won the George Polya Award from the Mathematical Association of America.

Cyclic sieving and cluster multicomplexes

Reiner, Stanton and White (2004) [10] proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu (2008) [2] 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 (2003) [6]. 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 seminoncrossing tableaux due to Pylyavskyy (2009) [9] as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky (2003) [5].

Games, puzzles, and computation

There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games. In the first half of this thesis. I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis. In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Konane, Cross Purposes, TipOver, and others. Some of these have been well-known open problems for some time. For other games, including Minesweeper, the Warehouseman's Problem, Sokoban, and Rush Hour, I either strengthen existing results, or provide new, simpler hardness proofs than the original proofs. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

Counting polygon dissections in the projective plane

For each value of k ⩾ 2 , we determine the number p n of ways of dissecting a polygon in the projective plane into n subpolygons with k + 1 sides each. In particular, if k = 2 we recover a result of Edelman and Reiner (1997) on the number of triangulations of the Möbius band having n labelled points on its boundary. We also solve the problem when the polygon is dissected into subpolygons of arbitrary size. In each case, the associated generating function ∑ p n z n is a rational function in z and the corresponding generating function of plane polygon dissections. Finally, we obtain asymptotic estimates for the number of dissections of various kinds, and determine probability limit laws for natural parameters associated to triangulations and dissections.

Combinatorial families enumerated by quasi-polynomials

We say that the sequence ( a n ) is quasi-polynomial in n if there exist polynomials P 0 , … , P s − 1 and an integer n 0 such that, for all n ⩾ n 0 , a n = P i ( n ) where i ≡ n ( mod s ) . We present several families of combinatorial objects with the following properties: Each family of objects depends on two or more parameters, and the number of isomorphism types of objects is quasi-polynomial in one of the parameters whenever the values of the remaining parameters are fixed to arbitrary constants. For each family we are able to translate the problem of counting isomorphism types of objects into the problem of counting integer points in a union of parametrized rational polytopes. The families of objects to which this approach is applicable include combinatorial designs, linear and unrestricted codes, and dissections of regular polygons.