Automatic Proofs for Formulae Enumerating Proper Polycubes

This video describes a general framework for computing formulae enumerating polycubes of size n which are proper in n-k dimensions (i.e., spanning all n-k dimensions), for a fixed value of k. (Such formulae are central in the literature of statistical physics in the study of percolation processes and collapse of branched polymers.) The implemented software re-affirmed the already-proven formulae for k <= 3, and proved rigorously, for the first time, the formula enumerating polycubes of size n that are proper in n-4 dimensions

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

High-dimensional holeyominoes

What is the maximum number of holes enclosed by a $d$-dimensional polyomino built of $n$ tiles? Represent this number by $f_d(n)$. Recent results show that $f_2(n)/n$ converges to $1/2$. We prove that for all $d \geq 2$ we have $f_d(n)/n \to (d-1)/d$ as $n$ goes to infinity. We also construct polyominoes in $d$-dimensional tori with the maximal possible number of holes per tile. In our proofs, we use metaphors from error-correcting codes and dynamical systems.Comment: 10 pages, 4 figure

Expansion in high dimension for the growth constants of lattice trees and lattice animals

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the expansion, making the present paper more self-contained with very little change to the structure of the original paper. Accepted for publication in Combinatorics Probability and Computin

A polyominoes-permutations injection and tree-like convex polyominoes

AbstractPlane polyominoes are edge-connected sets of cells on the orthogonal lattice Z2, considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [n], and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence an=6an−1−14an−2+16an−3−9an−4+2an−5, and compute the closed-form formula an=2n+2−(n3−n2+10n+4)/2

Detection of an anomalous cluster in a network

We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an "unusual behavior." Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. We also address surveillance settings where each sensor in the network collects information over time. The resulting model is similar, now with a time series attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal, and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The growth models used to represent the emerging cluster are quite general and, in particular, include cellular automata used in modeling epidemics. In both settings, we consider classes of clusters that are quite general, for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic, by far the most popular method in practice, achieves that same rate to within a logarithmic factor. Our results are not limited to the normal location model, but generalize to any one-parameter exponential family when the anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org