A187015 The number of different classes of 2-dimensional convex lattice polytopes having volume n/2 up to unimodular equivalence.

1, 2, 3, 7, 6, 13, 13, 27, 26, 44, 43, 83, 81, 122, 136, 208, 215, 317, 341, 490, 542, 710, 778, 1073, 1186, 1519, 1708, 2178, 2405, 3042, 3408, 4247, 4785, 5782, 6438, 7870, 8833, 10560, 11857, 14131, 15733, 18636, 20773, 24381, 27353, 31764, 35284, 41081, 45791, 52762

Offset: 1

Author

Jonathan Vos Post, Mar 01 2011

Keywords

• nonn

Comments

Lattice polytopes up to the equivalence relation used here are also called toric diagrams, see references below. - Andrey Zabolotskiy, May 10 2019

Liu & Zong give a(7) = 11, and others use their list, but their list lacks polygons No. 3 and 4 from Balletti's file 2-polytopes/v7.txt. - Andrey Zabolotskiy, Dec 28 2021

Links

• Günter Rote, Table of n, a(n) for n = 1..207
• Gabriele Balletti, Dataset of "small" lattice polytopes. Beware that the vertices are not always listed in sorted order around the polygon boundary (clockwise or counterclockwise).
• Gabriele Balletti, Enumeration of lattice polytopes by their volume, Discrete Comput. Geom., 65 (2021), 1087-1122; arXiv:1103.0103 [math.CO], 2018.
• Sebastián Franco, Yang-Hui He, Chuang Sun and Yan Xiao, A comprehensive survey of brane tilings, Int. J. Mod. Phys. A, 32 (2017), 1750142, arXiv:1702.03958 [hep-th], 2017.
• Heling Liu and Chuanming Zong, On the classification of convex lattice polytopes, Adv. Geom., 11 (2011), 711-729, arXiv:1103.0103 [math.MG], 2011. See table at p. 8.
• Günter Rote, Number of lattice polygons of area at most 103.5, classified by the number k of vertices, the number B of lattice points on edges, and the number I of interior lattice points.
• Yan Xiao, Quivers, Tilings and Branes, City, University of London, 2018. See Tables 3.2-3.7.

Crossrefs

Cf. A126587, A003051 (triangles only), A322343, A366409.

Programs

Python

# See the Python program for A366409.

Extensions

a(8) from Yan Xiao added by Andrey Zabolotskiy, May 10 2019

Name edited, a(7) corrected, a(9)-a(50) added using Balletti's data by Andrey Zabolotskiy, Dec 28 2021