Convex hulls of polyominoes

递增凸多边形

Ascending Convex Polyominoes

本文研究了Z-凸多边形的结构，引入基于西北和东北凸度的细化方法，将其分解为三个子类，并通过构造生成树得到生成函数的函数方程，从而获得显式代数生成函数和各子类的渐进行为。

Enumeration of symmetry classes of convex polyominoes in the square lattice

This paper concerns the enumeration of rotation-type and congruence-type convex polyominoes on the square lattice. These can be defined as orbits of the groups C4, of rotations, and D4, of symmetries of the square acting on (translation- type) polyominoes. In virtue of Burnside's Lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of C4 and D4. Using the Temperley--Bousquet-Melou methodology, we solve this problem and provide explicit or recursive formulas for their generating functions according to width, height and area. We …

The Number of Convex Polyominoes and the Generating Function of Jacobi Polynomials

Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$ \frac{m+n+mn}{m+n}{2m+2n\choose 2m}-\frac{2mn}{m+n}{m+n\choose m}^2. $$ We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials.

Enumeration of Symmetry Classes of Parallelogram Polyominoes

Parallelogram polyominoes are a subclass of convex polyominoes in the square lattice that has been studied extensively in the literature. Recently congruence classes of convex polyominoes with respect to rotations and reflections have been enumerated by counting orbits under the action of the dihedral group D4, of symmetries of the square, on (translation-type) convex polyominoes. Asymmetric convex polyominoes were also enumerated using Moebius inversion in the lattice of subgroups of D4. Here we extend these results to the subclass of parallelogram polyominos using a subgroup D2 of D4 which a…

A note on two identities arising from enumeration of convex polyominoes

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.

Polyominoes with nearly convex columns: A model with semidirected blocks

In most of today's exactly solved classes of polyominoes, either all members are convex (in some way), or all members are directed, or both. If the class is neither convex nor directed, the exact solution uses to be elusive. This paper is focused on polyominoes with hexagonal cells. Concretely, we deal with polyominoes whose columns can have either one or two connected components. Those polyominoes (unlike the well-explored column-convex polyominoes) cannot be exactly enumerated by any of the now existing methods. It is therefore appropriate to introduce additional restrictions, thus obtaining…

Convex polyominoes revisited: Enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees

The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that A) the average number of interior vertices over all convex polyominoes of perimeter $2n$ is asymptotic to $\frac{n^2}{12}+\frac{n\sqrt{n}}{3\sqrtπ} -\frac{(21π-16)n}{12π}.$ B) the average number of boundary vertices with degree two …

The area generating function for simple-2-column polyominoes with hexagonal cells

Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this chapter we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2,..., p connected components. Then column-convex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simplest generalization, namely to 2-column polyominoes, is unlikely to be solvable. We therefore define a class of polyominoes which interpolates between column-convex polygons and 2-column polyominoes. We d…

Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints

In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polyno…

The perimeter generating function for nondirected diagonally convex polyominoes

A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes, researchers also gave a look at diagonally convex polyominoes (DCPs), but noticed an awkward feature: when the last diagonal of a DCP is deleted, the remaining object is not always a polyomino.…

L-凸多联骨牌与避201上升序列的数值研究

A numerical study of L-convex polyominoes and 201-avoiding ascent sequences

本文研究了L-凸多联骨牌生成函数系数的渐近性态，并猜想出避201上升序列的D-有限代数解，同时展示了相关分析方法以期更广泛应用。

Regularity and Gorenstein property of the $L$-convex Polyominoes

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen--Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it.

Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice

Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explicit formulas or implicit functional equations for the generating series, which are convenient for computer algebra.

通过半周长的 k - 凸多联骨牌

$k$-Convex Polyominoes by Semi-perimeter

研究 k - 凸多联骨牌由半周长枚举的生成函数，通过分析生成的枚举数据得出推测解。

Linearly related polyominoes

We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the distributive lattices whose join-meet ideals are extremal Gorenstein or have a linear resolution.

反演关系、互反性与多联骨牌

Inversion relations, reciprocity and polyominoes

本文推导多联骨牌生成函数的自互反性质，并建立组合互反结果与统计力学反演关系的联系，从而完全确定若干凸多边形的各向异性周长生成函数。

使用正交凸多联骨牌对平面进行平移铺砌的不可判定性

Undecidability of Translational Tiling of the Plane with Orthogonally Convex Polyominoes

针对仅由几何形状决定的平面平移铺砌问题，本文摆脱以往极凹铺砌块构造，证明用一组 7 个正交凸多联骨牌铺砌平面仍不可判定。

Convex domino towers

We study convex domino towers using a classic dissection technique on polyominoes to find the generating function and an asymptotic approximation.

Convex hulls of face-vertex incident vectors of 3-colorable polytopes

The convex hulls of face-vertex incident vectors of 3-face-colorable convex polytopes are computed. It is found that every such convex hull is a $d$-polytope with $d+2$ or $d+3$ vertices. Utilizing Gale transform and Gale diagram, we calculate its combinatorial structure. Finally, a necessary and sufficient criterion for combinatorial equivalence of two such convex hulls is given.

Sumsets and the convex hull

We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times.

Convex Hull of Face Vectors of Colored Complexes

In this paper we verify a conjecture by Kozlov (Discrete Comput Geom 18 (1997) 421--431), which describes the convex hull of the set of face vectors of $r$-colorable complexes on $n$ vertices. As part of the proof we derive a generalization of Turán's graph theorem.

Towards the computation of the convex hull of a configuration from its corresponding separating matrix

In this paper, we cope with the following problem: compute the size of the convex hull of a configuration C, where the given data is the number of separating lines between any two points of the configuration (where the lines are generated by pairs of other points of the configuration). We give an algorithm for the case that the convex hull is of size 3, and a partial algorithm and some directions for the case that the convex hull is of size bigger than 3.

Automated counting of LEGO towers

H. N. V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple).

The Convex Hull of Parking Functions of Length $n$

Let $\mathcal{P}_n$ be the convex hull in $\mathbb{R}^n$ of all parking functions of length $n$. Stanley found the number of vertices and the number of facets of $\mathcal{P}_n$. Building upon these results, we determine the number of faces of arbitrary dimension, the volume, and the number of integer points of $\mathcal{P}_n$.

点生成的凸包格

Convex hull lattices point generated

本文研究了由点集生成的凸包格的有限性条件，通过给出点配置的概念，确定了平面上定义有限凸包格的点集（四个正则族和一个零散配置），并探讨了高维空间中的配置。

The $h$-polynomial and the rook polynomial of some polyominoes

Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots )/(1-t)^{\dim(\Bbbk[X])}$. For $k \in \mathbb{N}$, let $r_k$ be the number of configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2 < r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured characterization of thin polyominoes by Rinaldo and Romeo, \emph{J. Algebraic C…

Approximate Convex Hulls: sketching the convex hull using curvature

Convex hulls are fundamental objects in computational geometry. In moderate dimensions or for large numbers of vertices, computing the convex hull can be impractical due to the computational complexity of convex hull algorithms. In this article we approximate the convex hull in using a scalable algorithm which finds high curvature vertices with high probability. The algorithm is particularly effective for approximating convex hulls which have a relatively small number of extreme points.

Computing Convex Hulls in the Affine Building of SL_d

We describe an algorithm for computing the convex hull of a finite collection of points in the affine building of SL_d(K), for K a field with discrete valuation. These convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the convex hull as a tropical polytope.

Homothetic covering of convex hulls of compact convex sets

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $λ\in[0,1]$ such that $K$ can be covered by $m$ translates of $λK$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.

Computing the hull number in toll convexity

A walk $W$ between vertices $u$ and $v$ of a graph $G$ is called a {\em tolled walk between $u$ and $v$} if $u$, as well as $v$, has exactly one neighbour in $W$. A set $S \subseteq V(G)$ is {\em toll convex} if the vertices contained in any tolled walk between two vertices of $S$ are contained in $S$. The {\em toll convex hull of $S$} is the minimum toll convex set containing~$S$. The {\em toll hull number of $G$} is the minimum cardinality of a set $S$ such that the toll convex hull of $S$ is $V(G)$. The main contribution of this work is an algorithm for computing the toll hull number of a g…

凸包Thrackle

Convex Hull Thrackles

本文研究凸包thrackle这一线性thrackle的推广，证明点处于凸位置时凸包数不超过顶点数，一般情形下最多多一个，并给出凸包数不超过顶点数两倍的上界。

Rectilinear Convex Hull of Points in 3D

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ in general position, and let $RCH(P)$ be the rectilinear convex hull of $P$. In this paper we obtain an optimal $O(n\log n)$-time and $O(n)$-space algorithm to compute $RCH(P)$. We also obtain an efficient $O(n\log^2 n)$-time and $O(n\log n)$-space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of $P$ as we rotate $\mathbb R^3$ around the $z$-axis. Finally we study some properties of the rectilinear convex hulls of point sets in $\mathbb{R}^3$.

Convex hulls of superincreasing knapsacks and lexicographic orderings

We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the greedy solution to optimizing over this knapsack. We describe the convex hull of this $n$-dimensional set with $\mathcal{O}(n)$ facets. We also establish a distributive property by proving that the convex hull of $\le$- and $\ge$-type superincreasing knapsacks can be obtained by intersecting the convex hulls of $\le$- and $\ge$-sets taken individually. Our p…

Convex Hulls, Oracles, and Homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Given a set $Σ$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $ρ_1,ρ_2,...,ρ_m$, we show that the worst-case combinatorial complexity of the convex hull $CH_d(Σ)$ of $Σ$ is $Θ(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of spheres in $Σ$ with radius $ρ_i$. To prove the lower bound, we construct a set of $Θ(n_1+n_2)$ spheres in $\mathbb{E}^d$, with $d\ge{}3$ odd, where $n_i$ spheres have radius $ρ_i$, $i=1,2$, and $ρ_2\neρ_1$, such that their convex hull has combinatorial complexity $Ω(n_1n_2^{…

Expected mean width of the randomized integer convex hull

Let $K \in \R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K \cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(łK)-W((łK)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $ł\to \infty$.

Linear programs and convex hulls over fields of Puiseux fractions

We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful for computations in tropical geometry.

Tropical convex hulls of polyhedral sets

In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropically convex polyhedra in any dimension. %$\mathbb{R}^3/\mathbb {R}\mathbf{1}$. Finally we show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.

A note on Talagrand's convex hull concentration inequality

The paper reexamines an argument by Talagrand that leads to a remarkable exponential tail bound for the concentration of probability near a set. The main novelty is the replacement of a mysterious Calculus inequality by an application of Jensen's inequality.

On sumsets and convex hull

One classical result of Freimann gives the optimal lower bound for the cardinality of A+A if A is a d-dimensional finite set in the Euclidean d-space. Matolcsi and Ruzsa have recently generalized this lower bound to |A+kB| if B is d-dimensional, and A is contained in the convex hull of B. We characterize the equality case of the Matolcsi-Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.

The $2nd-$convex hull of every optimal rectilinear drawing of $K_{n}$ is a triangle

A rectilinear drawing of a graph $G$ is optimal if it has the smallest number of crossings among all rectilinear drawings of $G$. In this paper it is shown that for $n\geq 8$, the second convex hull of every optimal rectilinear drawing of the complete graph $K_n$ is a triangle.

The convex hull of a Banach-Saks set

A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{à}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.

How to Find the Convex Hull of All Integer Points in a Polyhedron?

We propose a cut-based algorithm for finding all vertices and all facets of the convex hull of all integer points of a polyhedron defined by a system of linear inequalities. Our algorithm DDM Cuts is based on the Gomory cuts and the dynamic version of the double description method. We describe the computer implementation of the algorithm and present the results of computational experiments comparing our algorithm with a naive one.