### Abstract

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice Z^{d}taking at least one step in every dimension. A polygon on Z^{d}is said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonsp_{n,d}of length 2nwithd(n)→∞ is asymptoticallyp_{n,d}~exp-2(2n-d)2n-1(2n-1)!(2πb(r)) ^{-1/2}r^{-2n+d}sinh^{d}r,wherer=r(n,d) is the unique solution ofrcothr=2n/d-1andb(r)=d(rcothr-r^{2}/sinh^{2}r). The convergence is uniform over alld≥ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1-d^{-1})^{2d}. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.

Original language | English |
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Pages (from-to) | 196-217 |

Number of pages | 22 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 80 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 1997 |

Externally published | Yes |

### Cite this

*Journal of Combinatorial Theory. Series A*,

*80*(2), 196-217. https://doi.org/10.1006/jcta.1997.2802

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*Journal of Combinatorial Theory. Series A*, vol. 80, no. 2, pp. 196-217. https://doi.org/10.1006/jcta.1997.2802

**Asymptotic Enumeration of Convex Polygons.** / Stark, Dudley; Wormald, Nicholas C.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Asymptotic Enumeration of Convex Polygons

AU - Stark, Dudley

AU - Wormald, Nicholas C.

PY - 1997/11

Y1 - 1997/11

N2 - A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice Zdtaking at least one step in every dimension. A polygon on Zdis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonspn,dof length 2nwithd(n)→∞ is asymptoticallypn,d~exp-2(2n-d)2n-1(2n-1)!(2πb(r)) -1/2r-2n+dsinhdr,wherer=r(n,d) is the unique solution ofrcothr=2n/d-1andb(r)=d(rcothr-r2/sinh2r). The convergence is uniform over alld≥ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1-d-1)2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.

AB - A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice Zdtaking at least one step in every dimension. A polygon on Zdis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonspn,dof length 2nwithd(n)→∞ is asymptoticallypn,d~exp-2(2n-d)2n-1(2n-1)!(2πb(r)) -1/2r-2n+dsinhdr,wherer=r(n,d) is the unique solution ofrcothr=2n/d-1andb(r)=d(rcothr-r2/sinh2r). The convergence is uniform over alld≥ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1-d-1)2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.

UR - http://www.scopus.com/inward/record.url?scp=0038879441&partnerID=8YFLogxK

U2 - 10.1006/jcta.1997.2802

DO - 10.1006/jcta.1997.2802

M3 - Article

AN - SCOPUS:0038879441

VL - 80

SP - 196

EP - 217

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -