A322343 -id:A322343 - cp's OEIS frontend

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

Jonathan Vos Post, Mar 01 2011

nonn

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

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.

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

Python
```
# See the Python program for A366409.
```

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

A322345 Maximal number of vertices of a convex lattice polygon containing n lattice points in its interior.

Original entry on oeis.org

4, 6, 6, 6, 8, 7, 8, 9, 8, 8, 10, 9, 9, 10, 10, 10, 10, 11, 10, 12, 12, 12, 11, 11, 12, 12, 12, 13, 12, 12, 13, 13, 13, 13, 14, 14, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 14, 15, 15, 15, 15, 15, 16, 15, 16, 15, 16, 16, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 16, 17, 17, 16, 17, 17

Offset: 0

Hugo Pfoertner, Dec 04 2018

nonn

This is an inverse of A063984 in the following sense: A063984(k) = min {n : a(n)>=k}. Thus a(n) grows roughly like const*n^(1/3). - Günter Rote, Sep 19 2023

Günter Rote, Table of n, a(n) for n = 0..200
Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column n_max in Table 1, p 512.
Wouter Castryck, Homepage. See the accompanying files for the above-referenced paper.
Günter Rote, Python program for this sequence and for A298562, (2023).
Günter Rote, Table of n, a(n) for n = 0..200 together with a corresponding a(n)-gon for each n, (2023).

Cf. A063984, A187015, A322343, A322346, A298562, A298755.

# See the Python program in the links section.

a(0) added by Andrey Zabolotskiy, Dec 29 2021
Name clarified by Günter Rote, Sep 18 2023
a(31) onwards from Günter Rote, Oct 01 2023

A371917 Number of inequivalent convex lattice polygons containing n lattice points (including points on the boundary).

Original entry on oeis.org

1, 3, 6, 13, 21, 41, 67, 111, 175, 286, 419, 643, 938, 1370, 1939, 2779, 3819, 5293, 7191, 9752, 12991, 17321, 22641, 29687, 38533, 49796, 63621, 81300, 102807, 129787, 162833, 203642, 252898, 313666, 386601, 475540, 582216, 710688, 863552, 1048176

Offset: 3

Justus Springer, Apr 12 2024

nonn

A322343 counts the polygons by their number of interior lattice points, excluding points on the boundary.

			For n = 3, the only polygon is the standard triangle with vertices (0,0), (1,0) and (0,1).
For n = 4, a(4) = 3 and the three polygons have vertex sets {(1,0),(0,1),(-1,-1)}, {(0,0),(2,0),(0,1)} and {(0,0),(1,0),(0,1),(1,1)}.

Justus Springer, Table of n, a(n) for n = 3..112
Martin Bohnert and Justus Springer, Classifying rational polygons with small denominator and few interior lattice points, arXiv:2410.17244 [math.CO], 2024. See p. 20.
R. J. Koelman, The number of moduli families of curves on toric surfaces, Dissertation (1991), Chapter 4.4.
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.
Justus Springer and M. Bohnert, Lattice polygons with at most 70 lattice points (1.0.0) [Data set], 2024.

Cf. A187015, A322343, A322344.

A322344 Number of equivalence classes of convex lattice polygons of genus n, restricting the count to those polygons that are interior to another polygon.

Original entry on oeis.org

16, 22, 63, 78, 122, 192, 239, 316, 508, 509, 700, 1044, 1113, 1429, 2052, 1962, 2651, 3543, 3638, 4594, 5996, 6364, 7922, 9692, 10208, 12727, 15431, 15918, 20354, 23873, 24677, 31593, 36529, 37302, 46034, 54454, 56278, 67020, 79606, 82549, 98188, 113752

Offset: 1

Hugo Pfoertner, Dec 04 2018

nonn

See Castryck article for an explanation how to check if a polygon is interior to another polygon by application of theorem 5 (Koelman 1991).

See A322343.

Justus Springer, Table of n, a(n) for n = 1..60
Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column N(1) in Table 1, p 512.
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.

Cf. A322343.

a(24) and a(30) corrected, a(31) onwards added by Justus Springer, Oct 26 2024

A322348 Maximal lattice width of a convex lattice polygon containing n lattice points in its interior ("of genus n").

Original entry on oeis.org

2, 3, 2, 4, 4, 4, 5, 4, 4, 5, 6, 5, 6, 6, 6, 7, 6, 6, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9

Offset: 0

Hugo Pfoertner, Dec 04 2018

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column lw_max in Table 1, p 512.

Cf. A322343, A322345, A322349.

a(0) added and name clarified by Andrey Zabolotskiy, Sep 21 2023

A322346 Maximal number of vertices of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

Original entry on oeis.org

6, 6, 6, 8, 7, 8, 9, 8, 8, 10, 9, 9, 10, 10, 10, 10, 11, 10, 12, 11, 12, 11, 11, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Hugo Pfoertner, Dec 04 2018

See A322343.

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column n_max(1) in Table 1, p 512.

Cf. A322343, A322345.

A322349 Maximal lattice width of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

Original entry on oeis.org

3, 2, 4, 4, 4, 5, 4, 4, 5, 6, 5, 6, 6, 6, 7, 6, 6, 7, 8, 7, 8, 8, 7, 8, 8, 8, 8, 9, 8, 9

Offset: 1

Hugo Pfoertner, Dec 04 2018

Sequence first differs from A322348 at n = 23.

See A322343.

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column lw_max(1) in Table 1, p 512.

Cf. A322343, A322348.

A322347 Minimal number of vertices of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

Original entry on oeis.org

3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 3, 3, 4, 4, 3, 4, 3, 4, 4, 4, 3, 4, 4, 3, 4, 4

Offset: 1

Hugo Pfoertner, Dec 04 2018

Without the restriction to polygons that are interior to another polygon, the minimal number of vertices is always 3.

See A322343.

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column n_min(1) in Table 1, p 512.

Cf. A322343.

A322350 Minimal number of points on the boundary of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

Original entry on oeis.org

3, 5, 5, 6, 7, 6, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 9, 9, 10, 10, 10, 10, 10, 11, 9, 10, 11, 11

Offset: 1

Hugo Pfoertner, Dec 04 2018

Without the restriction to polygons that are interior to another polygon, the minimal number of points on the boundary is always 3.

See A322343.

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column R_min(1) in Table 1, p 512.

Cf. A322343.

A374975 Number of equivalence classes of lattice polygons contained in a square of side length n but not in a square of side length n-1.

Original entry on oeis.org

2, 15, 131, 1369, 13842, 129185, 1104895, 8750964, 64714465, 450686225, 2976189422

Offset: 1

Justus Springer, Jul 26 2024

			For n = 1, only the square itself and the standard triangle are contained in the square, so a(1) = 2.

Gavin Brown and Alexander M. Kasprzyk, Small polygons and toric codes, J. Symbolic Comput. 51 (2013), pp. 55-62.
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.
Justus Springer and Martin Bohnert, Classifying rational polygons with small denominator and few interior lattice points, arxiv:2410.17244 [math.CO], 2024.
Justus Springer and Martin Bohnert, Lattice subpolygons of a square with given sidelength (1.0.0) [Data set], 2024.

Cf. A187015, A322343, A371917.

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

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A322345 Maximal number of vertices of a convex lattice polygon containing n lattice points in its interior.

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A371917 Number of inequivalent convex lattice polygons containing n lattice points (including points on the boundary).

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A322344 Number of equivalence classes of convex lattice polygons of genus n, restricting the count to those polygons that are interior to another polygon.

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A322348 Maximal lattice width of a convex lattice polygon containing n lattice points in its interior ("of genus n").

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A322346 Maximal number of vertices of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

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A322349 Maximal lattice width of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

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A322347 Minimal number of vertices of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

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A322350 Minimal number of points on the boundary of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

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A374975 Number of equivalence classes of lattice polygons contained in a square of side length n but not in a square of side length n-1.

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