相关论文: Enumeration of lozenge tilings of hexagons with cu…
We study lozenge tilings of a domain with partially free boundary. In particular, we consider a trapezoidal domain (half hexagon), s.t. the horizontal lozenges on the long side can intersect it anywhere to protrude halfway across. We show…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
This paper proposes a novel and simple algorithm of facet enumeration for convex polytopes. The complexity of the algorithm is discussed. The algorithm is implemented in Matlab. Some simple polytopes with known H-representations and…
We compute the number of rhombus tilings of a hexagon with sides $N,M,N,N,M,N$, which contain a fixed rhombus on the symmetry axis. A special case solves a problem posed by Jim Propp.
We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.
A general construction principle of inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of an expansion and a…
In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In…
The Heesch problem 'grades' polygons that fail to tile the plane in terms of the number of layers (or corollas) of copies of it that can be formed around a central unit. We study the different topology of ' walls', which we define to be…
Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three…
We introduce a computational origami problem which we call the segment folding problem: given a set of $n$ line-segments in the plane the aim is to make creases along all segments in the minimum number of folding steps. Note that a folding…
We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting…
We announce a solution to several enumeration problems in topology of surfaces. This includes an enumeration of homotopy classes of sections of locally trivial fiber bundles over surfaces and a computation of non-abelian 1-cohomology of…
We study the complexity of computing the projection of an arbitrary $d$-polytope along $k$ orthogonal vectors for various input and output forms. We show that if $d$ and $k$ are part of the input (i.e. not a constant) and we are interested…
We consider tromino tilings of $m\times n$ domino-deficient rectangles, where $3|(mn-2)$ and $m,n\geq0$, and characterize all cases of domino removal that admit such tilings, thereby settling the open problem posed by J. M. Ash and S.…
Which polygons admit two (or more) distinct lattice tilings of the plane? We call such polygons double tiles. It is well-known that a lattice tiling is always combinatorially isomorphic either to a grid of squares or to a grid of regular…
We introduce the problem of partitioning 2D regions (usually convex regions) into mutually congruent pieces ('tiles').
Mathematics has been used in the exploration and enumeration of juggling patterns. In the case when we catch and throw one ball at a time the number of possible juggling patterns is well-known. When we are allowed to catch and throw any…
We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding…
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations,…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…