Growth constants of polyominoes and lattice animals

edit · history · discussion · files · short url · statistical mechanics combinatorics generating function

Numbers

class

lattice

$d$

size

labelling

$\gamma$

site animals

square

cells

unlabelled:

4.062570 +/- 0.000008

comment: Klarner's constant. The series estimate is $4.062570(8)$ [1]. The rigorous bounds $4.00253<\gamma\leq4.5238$ are due to [4] and [6], respectively.

site animals

simple cubic

cells

unlabelled:

8.346638 +/- 0.000022

comment: The growth constant of fixed polycubes. Table 1 of [2] gives $\log\gamma=2.1218588(25)$ for the partially constrained fit. The number shown encloses its exponentiated uncertainty interval, with outward decimal rounding. A rigorous upper bound is $\gamma\leq9.3835$ [5].

bond animals

square

edges

unlabelled:

5.20789 +/- 0.00004 †

comment: The Monte Carlo estimate is $5.20789(4)$ in Section 5.2 of [2], with size equal to the number of chosen edges.

bond animals

simple cubic

edges

unlabelled:

10.61539 +/- 0.00006 †

comment: The Monte Carlo estimate is $10.61539(6)$ in Section 5.2 of [2], with size equal to the number of chosen edges.

Definition

The growth constant of a class of unlabelled lattice animals is $\gamma=1/\rho$, where $\rho$ is the radius of convergence of its ordinary generating function, counting site animals by cells and bond animals by edges [2].

Parameters

class

— animal class (site animals or bond animals)

lattice

— nearest-neighbour lattice (the hypercubic lattice $\mathbb Z^d$, using square for $d=2$ and sc for $d=3$)

$d$

— dimension ($d\geq2$; $d=2$ for square, $d=3$ for sc, and $d\geq4$ for hypercubic)

size

— size statistic (cells for site animals and edges for bond animals)

labelling

— labelling convention (unlabelled animals, with no labels on their vertices or edges)

Formulas

(1)

$\gamma=\lim_{n\to\infty}a_n^{1/n}=1/\rho$, where $a_n$ counts animals of size $n$ up to translation and $\rho$ is the radius of convergence of $\sum_{n\geq1}a_nz^n$ [2].

Comments

(2)

The lattice has vertex set $\mathbb Z^d$, with an undirected edge between two vertices whose difference is a signed coordinate unit vector. A site animal is a finite nonempty vertex set inducing a connected graph. Equivalently, put a unit cube at each occupied vertex and require connectivity across full faces; for $d=2$ these are polyominoes and for $d=3$ polycubes. A bond animal is a finite nonempty set of lattice edges whose union is connected. Different edge sets on the same occupied vertices are different bond animals [2].

(3)

Translations are identified, while rotations and reflections are distinguished. Every animal has weight one, and cavities and cycles are allowed. There is no distinguished root, preferred direction, confining wall or interaction energy. In particular, the bond-animal rows use the number of edges, not the number of vertices [2].

(4)

Each number is a numerical estimate with an explicit uncertainty, not a certified interval for the limiting constant. Three rows transcribe the cited papers directly; the simple-cubic site row converts a reported interval for $\log\gamma$ to one for $\gamma$. The uncertainties include the authors' assessment of extrapolation errors and are not rigorous error bounds.

(5)

The polyomino row retains Jensen's estimate $4.062570(8)$ [1]. Two newer estimates, $4.06256912(2)$ and $4.06256903(5)$, are reported in Section 6 of [3]; their quoted uncertainty ranges do not overlap. Both lie within the older uncertainty interval retained here, so this row does not claim the precision of either newer estimate.

(6)

Each choice of animal class and lattice specifies a counting problem. The table takes the same quantity $\gamma$, its exponential growth constant, in every case. The size statistic is part of that counting problem: occupied cells for site animals and chosen edges for bond animals.

References

[1]

Iwan Jensen, Enumerations of lattice animals and trees, Journal of Statistical Physics 102 (2001), 865–881. (arXiv) (doi)

[2]

Hsiao-Ping Hsu, Walter Nadler and Peter Grassberger, Simulations of lattice animals and trees, Journal of Physics A: Mathematical and General 38 (2005), 775–806. Table 1 and Section 5.2. (arXiv) (doi)

[3]

Gill Barequet and Gil Ben-Shachar, Counting polyominoes, revisited, Algorithmica 88 (2026), article 51, with the author-prepared corrigendum and addendum of October 2026. (arXiv) (doi)

[4]

Gill Barequet, Günter Rote and Mira Shalah, λ > 4: An improved lower bound on the growth constant of polyominoes, Communications of the ACM 59 (2016), 88–95. (doi)

[5]

Gill Barequet and Mira Shalah, Improved upper bounds on the growth constants of polyominoes and polycubes, 2019 preprint. (arXiv)

[6]

Vuong Bui, A convolutional approach to bounding the number of polyominoes, 2026 preprint, version 2. (arXiv)

[7]

E. J. Janse van Rensburg, The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles, second edition, Oxford University Press (2015), Section 2.1, p. 40.

Links

[8]

OEIS A001168: counts of fixed polyominoes

[9]

OEIS A001931: counts of fixed polycubes

Similar tables

Connective constants of lattices — growth constants for self-avoiding walks on the same lattices

Site and bond percolation thresholds of lattices — critical probabilities for independent site or bond occupation on these lattices

Growth constants of the classes of trees — growth constants for abstract tree classes, without a lattice embedding

Data properties

Entries are of type: real number

How well the digits are known: heuristic

Sources of data: [1], [2]

Table is complete: no (it holds both site and bond animals on the square and simple cubic lattices, the standard planar and spatial cases, using occupied cells for site animals and edges for bond animals; other lattices and dimensions are not included)

How they were obtained:

The attached generator reproduces published numerical estimates and their uncertainty intervals; it does not rerun the original Monte Carlo simulations. The polyomino row uses the wider series estimate of [1]. The simple-cubic site row uses the partially constrained fit in Table 1 of [2]. Its reported interval for $\log\gamma$ is exponentiated at 192 bits with Arb and rounded outwards. Repeating at 384 bits gives the same written interval; an independent 70-digit Decimal computation confirms containment of both transformed endpoints. This arithmetic verifies the conversion, not the source estimate.

Independent checks compare all four intervals with published estimates obtained by other methods, using the earlier estimates quoted by [2] for the cubic site and both bond rows. Exact enumeration gives the fixed-polyomino counts through 8 cells and the fixed-polycube counts through 6 cells, agreeing with [3], [8] and [9]. Two different enumerators agree on all bond counts through 4 edges on both lattices; the square counts also agree with [7]. Both enumerators first pass the one-dimensional control, whose counts are all 1. Two independent finite-series extrapolations of the polyomino counts lie within the retained interval; their control $a_n=7^n/n$ returns 7 exactly. The count at 67 cells uses the correction in [3]. These tests check conventions and numerical consistency; they do not prove the stated uncertainties.