Roland Bacher
After defining convex near-polygons, a formula enumerating the number of triangulations of such configurations is derived in terms of edge-polynomials. The paper describes also a transfer-matrix approach for computing quantities related to triangulations.
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Counting geodesics between surface triangulations
本研究考察曲面翻转图中两三角剖分间测地线数随距离的增长规律，证明拓扑足够复杂时为指数增长，其余情形为多项式增长。
We consider lattice walks in the plane starting at the origin, remaining in the first quadrant and made of West, South and North-East steps. In 1965, Germain Kreweras discovered a remarkably simple formula giving the number of these walks (with prescribed length and endpoint). Kreweras' proof was very involved and several alternative derivations have been proposed since then. But the elegant simplicity of the counting formula remained unexplained. We give the first purely combinatorial explanation of this formula. Our approach is based on a bijection between Kreweras walks and triangulations w…
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Ω(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Ω(n^{1/4})}$ Hamiltonian cycles.
Counting symmetric unimodular triangulations
研究平面扩张标准三角的三角剖分，聚焦具有固定对称轴的单模三角剖分，给出上下界并枚举小案例。
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations - a well-known open problem from computational geometry.
A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$). This paper gives formulas enumerating such triangulations (up to equivalence) on an oriented surface of given genus. {\em Une triangulation à un sommet} d'une surface orientée compacte $S$ de genre $g$ est un graphe $T\subset S$ qui a un unique sommet et dont toutes les faces (composantes connexes de $S\setminus T$) sont des triangles (incidentes à trois ar…
Let $P\subset\mathbb{R}^{2}$ be a set of $n$ points. In this paper we show two new algorithms, one to compute the number of triangulations of $P$, and one to compute the number of pseudo-triangulations of $P$. We show that our algorithms run in time $O^{*}(t(P))$ and $O^{*}(pt(P))$ respectively, where $t(P)$ and $pt(P)$ are the largest number of triangulation paths (T-paths) and pseudo-triangulations paths (PT-paths), respectively, that the algorithms encounter during their execution. Moreover, we show that $t(P) = O^{*}(9^{n})$, which is the first non-trivial bound on $t(P)$ to be known. Whil…
Counting cycles in planar triangulations
该研究围绕平面三角剖分图的指定长度环计数问题，推导了不同环长度范围、不同类型平面三角剖分下环数的上下界，完善了平面图环计数领域的理论结果。
We discuss the problem to count, or, more modestly, to estimate the number f(m,n) of unimodular triangulations of the planar grid of size $m\times n$. Among other tools, we employ recursions that allow one to compute the (huge) number of triangulations for small m and rather large n by dynamic programming; we show that this computation can be done in polynomial time if m is fixed, and present computational results from our implementation of this approach. We also present new upper and lower bounds for large m and n, and we report about results obtained from a computer simulation of the random …
Counting $k$-cycles in $5$-connected planar triangulations
该研究证明了5-连通平面三角剖分中5圈数上界为9n-50且紧，并给出k≥6时k圈数的渐近紧上界C(k)·n^{⌊k/3⌋}。
Given a set of $n$ points $S$ in the plane, a triangulation $T$ of $S$ is a maximal set of non-crossing segments with endpoints in $S$. We present an algorithm that computes the number of triangulations on a given set of $n$ points in time $n^{(11+ o(1))\sqrt{n} }$, significantly improving the previous best running time of $O(2^n n^2)$ by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size $O(\sqrt{n})$ of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers ("cactus graphs"). Bas…
Counting lattice triangulations: Fredholm equations in combinatorics
针对$m×n$矩形本原格点三角剖分数量$f(m,n)$，计算得$m=2$时极限精确值为$(611+\sqrt{73})/36$，$m=3$时用Fredholm积分方程表示极限并给出多项式时间高精度计算算法。
Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations of $P$, among others. In this paper we develop a general technique for computing the number of crossing-free structures of an input set $P$. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of $P$. The running time of our algorithms is upper bounded by $n^{O(k)}$, where $k$ is the number o…
Combinatorial zeta functions counting triangles
本文计算了n维流形三角剖分(n-1)-骨架中计数测地路径的组合Zeta函数特殊值，利用随机游走揭示其拓扑意义，恢复了流形的Betti数与环绕数。
Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a class of configurations for which this problem can be solved to some extent. Loosely speaking, a nearly convex polygon is an infinitesimal perturbation of a weakly convex polygon (a convex polygon with edges subdivided by additional points). Our main result shows that the triangulation polynomial, enumerating all triangulations of a nearly convex polygon, is def…
Three--dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the notion of two--dimensional $2p$--angulations to three dimensions in a way which is suitable for combinatorics and enumeration. In particular, universality classes of three--dimensional triangulations can be investigated within this framework. Here we study colored triangulations obtained by gluing octahedra. Those which maximize the number of edges at fixed …
The triangulations of point configurations which decompose as a free sum are classified in terms of the triangulations of the summands. The methods employ two new kinds of partially ordered sets to be associated with any triangulation of a point set with one point marked, the "web of stars" and the "stabbing poset". Triangulations of smooth Fano polytopes are discussed as a case study.
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms an…
We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.
We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size $n$ decorated by an Ising configuration with a weight proportional to the energy of this configuration. To do so, we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition. In particular, we show that these partition functions all have the same phase transition at the same critical temperature. Some properties of the limiting object -- called the Infinite Ising Planar Triangulation -- are derived, in…
Symmetric lexicographic symmetric-subset reverse search for the enumeration of circuits, cocircuits, and triangulations up to symmetry
研究利用对称字典序对称子集反向搜索框架枚举有限集对称可行子集，针对余电路、电路和三角剖分应用提出新方法，经C++实现后大幅提速，还计算出多个新数量并发现相关实例问题。
We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has Omega(1.5028^n) convex polygons, and prove an O(1.62^n) upper bound in the worst case. If the triangulation is fat (every triangle has its angles lower-bounded by a constant delta>0), then there can be only polynomially many. We also consider the problem of counting convex outerplanar polygons (i.e., they contain no vertices of the triangulation in their inte…
A normal (respectively, graded normal) vector configuration $A$ defines the toric ideal $I_A$ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when $A$ is normal and graded, $I_A$ is generated in degree at most the dimension of $I_A$. Based on this, Sturmfels asked if these properties extend to initial ideals -- when $A$ is normal, is there an initial ideal of $I_A$ that is Cohen-Macaulay, and when $A$ is normal and graded, does $I_A$ have a Gröbner basis generated in degree at most $dim(I_A)$ ? In this paper, we answer both questions positive…
We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. As a consequence, random counterparts are presented as well.
Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad-hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.
Cross-ratio degrees and triangulations
本文研究交比度的组合结构，针对由 n 边形三角剖分索引的一类交比度给出简洁闭式公式，并揭示其与 M_{0,n} 实轨迹几何及正几何的联系。
New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the Temperley-Lieb chain, for periodic boundary conditions and so-called ``identified connectivities'', up to size $2n=22$.
Origami, where two-dimensional sheets are folded into complex structures, is proving to be rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider \emph{flat origami}, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases made by such foldings, called a \emph{MV assignment} of the crease pattern. We establish a method to, given a flat-foldable crease pattern $C$ under certain conditions, create a planar graph $C^*$ whose 3-colorings are in one-to-one c…
We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $\mathbb{C}$, and the space of unordered $n$-tuples of linearly independent lines in $\mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations given by a character polynomial, we compute $H^i(Y_n; W_n)$ for all $i$ and all $n$ in terms of double generating functions. One consequence of the computation is a new recurrence phenomenon: the stable twisted Betti numbers $\lim_{n\to\infty}\dim H^i(Y_n; W_n)$ are linearly recurren…
Counting point configurations in projective space
本文提出了一种在射影空间中计数具有固定相对位置点配置的方法，并建立了与不同维度射影空间相关的递推关系。
A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of a particular type of k-uniform hypergraph. In this paper we introduce a fully polynomial randomised approximation scheme (FPRAS) which counts the number of k-stable c-coloured Candy Crush grids of a given size (m, n) for certain values of c and k. We implemented this algorithm on Matlab, and found that in a Candy Crush grid with7 available colours there are approximately 4.3*10^61 3-stable colourings. (Note that, typical Candy Crush games are played with 6 colours and our FPRAS is not guaranteed to work in expected polynomi…
Parallel Diffusion is a variant of Chip-Firing introduced in 2018 by Duffy et al. In Parallel Diffusion, chips move from places of high concentration to places of low concentration through a discrete-time process. At each time step, every vertex sends a chip to each of its poorer neighbours, allowing for some vertices to perhaps fall into debt (represented by negative stack sizes). In their recent paper, Long and Narayanan proved a conjecture from the original paper by Duffy et al. that every Parallel Diffusion process eventually, after some pre-period, exhibits periodic behaviour. With this r…
Counting LEGO configurations
本文研究了计算由平行w×1乐高积木组成的结构数量问题，分析了二维和三维结构的配置增长规律，并提出多项式形式的序列表达式。
One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erdős, Kleitman and Rothschild, who showed that the number of $K_r$-free graphs on $n$ vertices is $2^{\text{ex}(n,K_r)+o(n^2)}$. Their work was later extended to forbidding graphs as induced subgraphs by Prömel and Steger. Here, we consider one of the most basic counting problems for $3$-graphs. Let $E_1$ be the $3$-graph with $4$ vertices and $1$ edge. What is the number of induced …
Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.
On the threshold for triangulations inside convex polygons
本文研究凸多边形内部随机加边出现三角剖分的临界阈值，通过耳切算法和赌徒破产问题证明 $p_c<1/2$，表明大凸多边形内大多数边配置包含三角剖分。
We describe a new conjecture involving Fully Packed Loop counting which relates recent observations of Thapper to formulae in the Temperley--Lieb model of loops, and how it implies the Razumov--Stroganov conjecture.
Counting subsets of integers free of arithmetic configurations
本文改进了关于无等差数列子集计数的问题，证明了对于k≥5，无限多个n值下计数结果为2^{r_k(n)(1+o(1))}，并为k≥3的所有n提供了2^{O(r_k(n))}的计数结果。
Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are related to certain configurations of vicious random walkers and to the enumeration of plane partitions. The combinatorial meaning of the average of the characteristic polynomial of random Hermitian and Wishart matrices is also investigated, and consequently several simple universality results are derived.
We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. the expansion coefficients when multiplying a Schubert class repeatedly with different Chern classes. This allows one to derive sum rules for Gromov-Witten invariants.
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