Formulae and growth rates of high-dimensional polycubes

  title={Formulae and growth rates of high-dimensional polycubes},
  author={Ronnie Barequet and Gill Barequet and G{\"u}nter Rote},

The growth rate of high-dimensional tree polycubes

Proper n-Cell Polycubes in n - 3 Dimensions

A formula is proved for the number of polycubes of size n that are proper in (n - 3) dimensions, which is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional.

Counting n-cell polycubes proper in n-k dimensions

Polycubes with Small Perimeter Defect

A few formulae enumerating polycubes with a fixed deviation from the maximum possible perimeter are presented.

Formulae and Growth Rates of Animals on Cubical and Triangular Lattices

A polyomino of size n consists of n squares joined along their edges. A popular example is the computer game Tetris, which features polyominoes of size 4. A d-dimensional polycube of size n is a

The Perimeter of Proper Polycubes

Formulas are derived for the number of polycubes of size $n$ and perimeter $t$ that are proper in $n-1 and$n-2$ dimensions that complement computer based enumerations of perimeter polynomials in percolation problems.

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

The method for the best known upper bound on A2(n) is revisited and extended and the number of d-dimensional polycubes with n cubes is defined.

C O ] 1 0 M ay 2 01 7 The Perimeter of Proper Polycubes

We derive formulas for the number of polycubes of size n and perimeter t that are proper in n − 1 and n − 2 dimensions. These formulas complement computer based enumerations of perimeter polynomials

Automatic Proofs for Formulae Enumerating Proper Polycubes



Counting d-Dimensional Polycubes and nonrectangular Planar polyominoes

This paper describes a generalization of Redelmeier’s algorithm for counting two-dimensional rectangular polyominoes and computed the number of distinct 3-D polycubes of size 18, which is the first tabulation of this value.

Counting polycubes without the dimensionality curse

Counting Polyominoes on Twisted Cylinders

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We

Percolation processes in d-dimensions

Series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented. Numerical evidence for the existence of a critical dimension for the cluster

Counting Polyominoes: A Parallel Implementation for Cluster Computing

  • I. Jensen
  • Computer Science, Mathematics
    International Conference on Computational Science
  • 2003
This work has developed a parallel algorithm for the enumeration of polyominoes, which are connected sets of lattice cells joined at an edge, and implements the finite-lattice method and associated transfer-matrix calculations in a very efficient parallel setup.

Cell Growth Problems

  • D. Klarner
  • Mathematics
    Canadian Journal of Mathematics
  • 1967
The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit


The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions

It is proved existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on Z d, and the method uses the lace expansion.

Counting polyominoes: Yet another attack

A pattern theorem for lattice clusters

We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration (“pattern”) of sites and bonds can