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{"Title": "Growth constants of polyominoes and lattice animals", "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 CITE{HNG}.", "Keywords": ["Klarner constant", "Klarner's constant", "polyomino growth constant", "polycube growth constant", "site animals", "bond animals", "fixed polyominoes", "fixed polycubes", "lattice animal enumeration"], "Parameters": {"class": {"type": "Symbolic", "title": "animal class", "display": "class", "constraints": "site animals or bond animals", "values": {"site": "site animals", "bond": "bond animals"}}, "lattice": {"type": "Symbolic", "title": "nearest-neighbour lattice", "display": "lattice", "constraints": "the hypercubic lattice $\\mathbb Z^d$, using square for $d=2$ and sc for $d=3$", "values": {"square": "square", "sc": "simple cubic", "hypercubic": "hypercubic $\\mathbb Z^d$"}}, "d": {"type": "Z", "title": "dimension", "display": "$d$", "constraints": "$d\\geq2$; $d=2$ for square, $d=3$ for sc, and $d\\geq4$ for hypercubic"}, "size": {"type": "Symbolic", "title": "size statistic", "display": "size", "constraints": "cells for site animals and edges for bond animals", "values": {"cells": "number of occupied cells", "edges": "number of chosen edges"}}, "labelling": {"type": "Symbolic", "title": "labelling convention", "display": "labelling", "constraints": "unlabelled animals, with no labels on their vertices or edges", "values": {"unlabelled": "unlabelled"}}}, "Comments": {"objects": "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 CITE{HNG}.", "conventions": "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 CITE{HNG}.", "estimates": "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.", "polyomino-precision": "The polyomino row retains Jensen's estimate $4.062570(8)$ CITE{Jensen}. Two newer estimates, $4.06256912(2)$ and $4.06256903(5)$, are reported in Section 6 of CITE{BB}; 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.", "family": "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."}, "Similar tables": [{"table": "HREF{Connective_constants_of_lattices}", "relation": "growth constants for self-avoiding walks on the same lattices"}, {"table": "HREF{Site_and_bond_percolation_thresholds_of_lattices}", "relation": "critical probabilities for independent site or bond occupation on these lattices"}, {"table": "HREF{Growth_constants_of_the_classes_of_trees}", "relation": "growth constants for abstract tree classes, without a lattice embedding"}], "Links": {"OEISpolyomino": {"title": "OEIS A001168: counts of fixed polyominoes", "url": "https://oeis.org/A001168"}, "OEISpolycube": {"title": "OEIS A001931: counts of fixed polycubes", "url": "https://oeis.org/A001931"}}, "References": {"Jensen": {"bib": "Iwan Jensen, Enumerations of lattice animals and trees, Journal of Statistical Physics 102 (2001), 865\u2013881.", "arxiv": "cond-mat/0007239", "doi": "10.1023/A:1004855020556"}, "HNG": {"bib": "Hsiao-Ping Hsu, Walter Nadler and Peter Grassberger, Simulations of lattice animals and trees, Journal of Physics A: Mathematical and General 38 (2005), 775\u2013806. Table 1 and Section 5.2.", "doi": "10.1088/0305-4470/38/4/001", "arxiv": "cond-mat/0408061"}, "BB": {"bib": "Gill Barequet and Gil Ben-Shachar, Counting polyominoes, revisited, Algorithmica 88 (2026), article 51, with the author-prepared corrigendum and addendum of October 2026.", "doi": "10.1007/s00453-026-01392-6", "arxiv": "2610.10632"}, "BRS": {"bib": "Gill Barequet, G\u00fcnter Rote and Mira Shalah, \u03bb > 4: An improved lower bound on the growth constant of polyominoes, Communications of the ACM 59 (2016), 88\u201395.", "doi": "10.1145/2851485"}, "BS": {"bib": "Gill Barequet and Mira Shalah, Improved upper bounds on the growth constants of polyominoes and polycubes, 2019 preprint.", "arxiv": "1906.11447"}, "Bui": {"bib": "Vuong Bui, A convolutional approach to bounding the number of polyominoes, 2026 preprint, version 2.", "arxiv": "2511.00461"}, "Rensburg": {"bib": "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."}}, "Tags": ["combinatorics", "generating function", "statistical mechanics"], "Data properties": {"type": "R", "rigour": "heuristic", "sources": ["CITE{Jensen}", "CITE{HNG}"], "complete": "no", "complete-note": "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", "rigour details": "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 CITE{Jensen}. The simple-cubic site row uses the partially constrained fit in Table 1 of CITE{HNG}. 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.\n\nIndependent checks compare all four intervals with published estimates obtained by other methods, using the earlier estimates quoted by CITE{HNG} 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 CITE{BB}, CITE{OEISpolyomino} and CITE{OEISpolycube}. Two different enumerators agree on all bond counts through 4 edges on both lattices; the square counts also agree with CITE{Rensburg}. 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 CITE{BB}. These tests check conventions and numerical consistency; they do not prove the stated uncertainties."}, "Display properties": {"number-header": "$\\gamma$"}, "Numbers": {"site": {"square": {"2": {"cells": {"unlabelled": {"number": "4.062570 +/- 0.000008", "comment": "Klarner's constant. The series estimate is $4.062570(8)$ CITE{Jensen}. The rigorous bounds $4.00253<\\gamma\\leq4.5238$ are due to CITE{BRS} and CITE{Bui}, respectively."}}}}, "sc": {"3": {"cells": {"unlabelled": {"number": "8.346638 +/- 0.000022", "comment": "The growth constant of fixed polycubes. Table 1 of CITE{HNG} 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 $\\gamma\\leq9.3835$ CITE{BS}."}}}}}, "bond": {"square": {"2": {"edges": {"unlabelled": {"number": "5.20789 +/- 0.00004", "comment": "The Monte Carlo estimate $5.20789(4)$ in Section 5.2 of CITE{HNG}, with size equal to the number of chosen edges."}}}}, "sc": {"3": {"edges": {"unlabelled": {"number": "10.61539 +/- 0.00006", "comment": "The Monte Carlo estimate $10.61539(6)$ in Section 5.2 of CITE{HNG}, with size equal to the number of chosen edges."}}}}}}, "Formulas": {"growth": "$\\gamma=\\lim_{n\\to\\infty}a_n^{1/n}=1/\\rho$, where $a_n$ counts animals of size $n$ up to translation and $\\rho$ is the radius of convergence of $\\sum_{n\\geq1}a_nz^n$ CITE{HNG}."}}