Counting Lattice Animals in High Dimensions

We present an implementation of Redelemeier's algorithm for the enumeration of lattice animals in high dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter polynomials and series expansion coefficients in $d$-dimensional hypercubic lattices for $3 \leq d\leq 10$. From the data we compute formulas for perimeter polynomials for lattice animals of size $n\leq 11$ in arbitrary dimension $d$. When amended by combinatorial arguments, the new data suffices to yield explicit formulas for the number of lattice animals of size $n\leq 14$ and arbitrary $d$. We also use the enumeration data to compute numerical estimates for growth rates and exponents in high dimensions that agree very well with Monte Carlo simulations and recent predictions from field theory.Comment: 18 pages, 7 figures, 6 tables; journal versio

A new mathematical model for tiling finite regions of the plane with polyominoes

We present a new mathematical model for tiling finite subsets of $\mathbb{Z}^2$ using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear programming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Counting polycubes without the dimensionality curse

Abstractd-dimensional polycubes are the generalization of planar polyominoes to higher dimensions. That is, a d-D polycube of size n is a connected set of n cells of a d-dimensional hypercubic lattice, where connectivity is through (d−1)-dimensional faces of the cells. Computing Ad(n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier [D.H. Redelmeier, Counting polyominoes: Yet another attack, Discrete Math. 36 (1981) 191–203]. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present an improved version of the same method, whose order of memory consumption is a (very low) polynomial in both n and d. We also describe how we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously