A379894 Number of rational polygons of denominator at most n having exactly one lattice point in their interior and primitive vertices, up to equivalence.

16, 505, 48032, 1741603, 154233886, 2444400116

Offset: 1

Author

Justus Springer, Jan 05 2025

Keywords

• nonn
• more

Comments

A rational polygon P of denominator d is said to have primitive vertices, if the lattice polygon d*P has primitive vertices.

A379887 counts the polygons without the condition that the vertices are primitive. Both are in Classification 5.6 of the article by Bohnert and Springer.

a(n) is also the number of isomorphism classes of 1/n-log canonical toric del Pezzo surfaces, see the article by Hättig, Hausen, Hafner and Springer.

An algorithm to compute a(n) was given by Timo Hummel in his dissertation. His final number for n = 3 (given in Corollary 12.2) was however slightly off.

Examples

			For n = 1, there are 16 lattice polygons with exactly one interior lattice point, which are the 16 reflexive lattice polygons.
		

Links

• Martin Bohnert and Justus Springer, Classifying rational polygons with small denominator and few interior lattice points, arXiv:2410.17244 [math.CO], 2024.
• Martin Bohnert and Justus Springer, Rational polygons with exactly one interior lattice point [Data set]. Zenodo.
• Daniel Hättig, Jürgen Hausen, and Justus Springer, Classifying log del Pezzo surfaces with torus action, arXiv:2302.03095 [math.AG], 2023.
• Daniel Hättig, Lattice Polygons and Surfaces with Torus Action, Dissertation (2023).
• Timo Hummel, Automorphisms of rational projective K*-surfaces, Dissertation (2021).
• Justus Springer, RationalPolygons.jl (Version 1.1.0) [Computer software], 2024.

Crossrefs

Cf. A379887, A322343, A141682, A145581, A371917.