Related papers: Counting lattice triangulations: Fredholm equation…

Let $f(m,n)$ be the number of primitive lattice triangulations of an $m \times n$ rectangle. We express the limits $\lim_n f(m,n)^{1/n}$ for $m = 4$ and $m=5$ in terms of certain systems of Fredholm integral equations on generating…

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…

We prove an exponential upper bound for the number $f(m,n)$ of all maximal triangulations of the $m\times n$ grid: \[ f(m,n) < 2^{3mn}. \] In particular, this improves a result of S. Yu. Orevkov (1999).

We study the exact counting problem for all lattice rectangles contained in the square $[0,n)\times[0,n)$, including non-axis-parallel ones. Starting from the standard parametrization by a primitive direction $(u,v)$ and two side lengths,…

We give a simple formula for the signature of a foldable triangulation of a lattice polygon in terms of its boundary. This yields lower bounds on the number of real roots of certain of systems of polynomial equations known as "Wronski…

The problem of counting the different ways of folding the planar triangular lattice is shown to be equivalent to that of counting the possible 3-colorings of its bonds, a dual version of the 3-coloring problem of the hexagonal lattice…

We estimate the lattice sums arising in the context of the integer point counting in polyhedra.

In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and $n$ additional vertices. In this note, we study how many ways a combinatorial triangulation…

In this paper we use an iterative algorithm for solving Fredholm equations of the first kind. The basic algorithm is known and is based on an EM algorithm when involved functions are non-negative and integrable. With this algorithm we…

In this work, we consider a number of problems defined on the triangular lattice with $n$ rows, which we will denote as $T_n$. Define a \textit{proper coloring} to be an assignment of colors to the points of $T_n$ such that no three points…

This paper is dedicated to a lattice analog to the classical ``sum of interior angles of a polygon theorem''. In 2008, the first formula expressing conditions on the geometric continued fractions for lattice angles of triangles was derived,…

Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special…

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…

The number of triangulations of a planar n point set is known to be $c^n$, where the base $c$ lies between $2.43$ and $30.$ The fastest known algorithm for counting triangulations of a planar n point set runs in $O^*(2^n)$ time. The fastest…

In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of…

We use the residue theorem to derive an expression for the number of lattice oints in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart…

We enumerate rooted triangulations of a sphere with multiple holes by the total number of edges and the length of each boundary component. The proof relies on a combinatorial identity due to W.T. Tutte.

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects…

We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration…

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one of which…