Related papers: Polymetric brick wall patterns and two-dimensional…
We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions…
Building blocks and tiles are an excellent way of learning about geometry and mathematics in general. There are several versions of tiles that are either snapped together or connected with magnets that can be used to introduce topics like…
Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled…
We consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds…
A new construction for moderate density parity-check (MDPC) codes using finite geometry is proposed. We design a parity-check matrix for this family of binary codes as the concatenation of two matrices: the incidence matrix between points…
A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give…
A method for efficiently constructing polar codes is presented and analyzed. Although polar codes are explicitly defined, straightforward construction is intractable since the resulting polar bit-channels have an output alphabet that grows…
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 study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of…
We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). We show that a maze of…
The number of distinguishable inherent structures of a liquid is the key component to understanding the thermodynamics of glass formers. In the case of hard potential systems such as hard discs, spheres and ellipsoids, an inherent structure…
The aim of this paper is to derive explicit formulas for two distinct values. The first is the total number of symmetric peaks in a set partition of $[n]$ with exactly $k$ blocks, and the second one is the total number of non-symmetric…
The generating function of reverse plane partitions of a fixed shape factors into a product featuring the hook-lengths of this shape. This result, which was first obtained by Stanley, can be explained bijectively using the Hillman-Grassl…
We study a model of two polymers confined to a slit with sticky walls. More precisely, we find and analyse the exact solution of two directed friendly walks in such a geometry on the square lattice. We compare the infinite slit limit, in…
Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…
We consider a planar graph $G$ in which the edges have nonnegative integer lengths such that the length of every cycle of $G$ is even, and three faces are distinguished, called holes in $G$. It is known that there exists a packing of cuts…
Diverse planning is the problem of finding multiple plans for a given problem specification, which is at the core of many real-world applications. For example, diverse planning is a critical piece for the efficiency of plan recognition…
Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather…
A $d$-dimensional brick is a set $I_1\times \cdots \times I_d$ where each $I_i$ is an interval. Given a brick $B$, a brick partition of $B$ is a partition of $B$ into bricks. A brick partition $\mathcal{P}_d$ of a $d$-dimensional brick is…
We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given,…