3D Alcoved Data

There are 6 types of 3-dimesional alcoved polytopes up to symmetry. We follow the order of types listed on this webpage accompanying the paper “Multivariate volume, Ehrhart, and h⁎-polynomials of polytropes“ by Brandenburg, Elia, and Zhang (2023), (arXiv version). Their website shows the unweighted Ehrhart and h* polynomials for 2 to 4 dimensional alcoved polytopes.

Multivariate Ehrhart Functions of 3D Alcoved Polytopes

Multivariate Ehrhart Functions of 3D Alcoved Polytopes

Weight Type 1 Type 2 Type 3 Type 4 Type 5 Type 6
Ehrhart\(h^*\) Ehrhart\(h^*\) Ehrhart\(h^*\) Ehrhart\(h^*\) Ehrhart\(h^*\) Ehrhart\(h^*\)
1 ehr_1_1 h*_1_1 ehr_2_1 h*_2_1 ehr_3_1 h*_3_1 ehr_4_1 h*_4_1 ehr_5_1 h*_5_1 ehr_6_1 h*_6_1
\(x_1\) ehr_1_x1 h*_1_x1 ehr_2_x1 h*_2_x1 ehr_3_x1 h*_3_x1 ehr_4_x1 h*_4_x1 ehr_5_x1 h*_5_x1 ehr_6_x1 h*_6_x1
\(x_2\) ehr_1_x2 h*_1_x2 ehr_2_x2 h*_2_x2 ehr_3_x2 h*_3_x2 ehr_4_x2 h*_4_x2 ehr_5_x2 h*_5_x2 ehr_6_x2 h*_6_x2
\(x_3\) ehr_1_x3 h*_1_x3 ehr_2_x3 h*_2_x3 ehr_3_x3 h*_3_x3 ehr_4_x3 h*_4_x3 ehr_5_x3 h*_5_x3 ehr_6_x3 h*_6_x3
\(x_1x_2\) ehr_1_x1x2 h*_1_x1x2 ehr_2_x1x2 h*_2_x1x2 ehr_3_x1x2 h*_3_x1x2 ehr_4_x1x2 h*_4_x1x2 ehr_5_x1x2 h*_5_x1x2 ehr_6_x1x2 h*_6_x1x2
\(x_1x_3\) ehr_1_x1x3 h*_1_x1x3 ehr_2_x1x3 h*_2_x1x3 ehr_3_x1x3 h*_3_x1x3 ehr_4_x1x3 h*_4_x1x3 ehr_5_x1x3 h*_5_x1x3 ehr_6_x1x3 h*_6_x1x3
\(x_1^2\) ehr_1_x1^2 h*_1_x1^2 ehr_2_x1^2 h*_2_x1^2 ehr_3_x1^2 h*_3_x1^2 ehr_4_x1^2 h*_4_x1^2 ehr_5_x1^2 h*_5_x1^2 ehr_6_x1^2 h*_6_x1^2
\(x_2x_3\) ehr_1_x2x3 h*_1_x2x3 ehr_2_x2x3 h*_2_x2x3 ehr_3_x2x3 h*_3_x2x3 ehr_4_x2x3 h*_4_x2x3 ehr_5_x2x3 h*_5_x2x3 ehr_6_x2x3 h*_6_x2x3
\(x_2^2\) ehr_1_x2^2 h*_1_x2^2 ehr_2_x2^2 h*_2_x2^2 ehr_3_x2^2 h*_3_x2^2 ehr_4_x2^2 h*_4_x2^2 ehr_5_x2^2 h*_5_x2^2 ehr_6_x2^2 h*_6_x2^2
\(x_3^2\) ehr_1_x3^2 h*_1_x3^2 ehr_2_x3^2 h*_2_x3^2 ehr_3_x3^2 h*_3_x3^2 ehr_4_x3^2 h*_4_x3^2 ehr_5_x3^2 h*_5_x3^2 ehr_6_x3^2 h*_6_x3^2
\(x_1x_2x_3\) ehr_1_x1x2x3 h*_1_x1x2x3 ehr_2_x1x2x3 h*_2_x1x2x3 ehr_3_x1x2x3 h*_3_x1x2x3 ehr_4_x1x2x3 h*_4_x1x2x3 ehr_5_x1x2x3 h*_5_x1x2x3 ehr_6_x1x2x3 h*_6_x1x2x3
\(x_1x_2^2\) ehr_1_x1x2^2 h*_1_x1x2^2 ehr_2_x1x2^2 h*_2_x1x2^2 ehr_3_x1x2^2 h*_3_x1x2^2 ehr_4_x1x2^2 h*_4_x1x2^2 ehr_5_x1x2^2 h*_5_x1x2^2 ehr_6_x1x2^2 h*_6_x1x2^2
\(x_1x_3^2\) ehr_1_x1x3^2 h*_1_x1x3^2 ehr_2_x1x3^2 h*_2_x1x3^2 ehr_3_x1x3^2 h*_3_x1x3^2 ehr_4_x1x3^2 h*_4_x1x3^2 ehr_5_x1x3^2 h*_5_x1x3^2 ehr_6_x1x3^2 h*_6_x1x3^2
\(x_1^2x_2\) ehr_1_x1^2×2 h*_1_x1^2×2 ehr_2_x1^2×2 h*_2_x1^2×2 ehr_3_x1^2×2 h*_3_x1^2×2 ehr_4_x1^2×2 h*_4_x1^2×2 ehr_5_x1^2×2 h*_5_x1^2×2 ehr_6_x1^2×2 h*_6_x1^2×2
\(x_1^2x_3\) ehr_1_x1^2×3 h*_1_x1^2×3 ehr_2_x1^2×3 h*_2_x1^2×3 ehr_3_x1^2×3 h*_3_x1^2×3 ehr_4_x1^2×3 h*_4_x1^2×3 ehr_5_x1^2×3 h*_5_x1^2×3 ehr_6_x1^2×3 h*_6_x1^2×3
\(x_1^3\) ehr_1_x1^3 h*_1_x1^3 ehr_2_x1^3 h*_2_x1^3 ehr_3_x1^3 h*_3_x1^3 ehr_4_x1^3 h*_4_x1^3 ehr_5_x1^3 h*_5_x1^3 ehr_6_x1^3 h*_6_x1^3
\(x_2x_3^2\) ehr_1_x2x3^2 h*_1_x2x3^2 ehr_2_x2x3^2 h*_2_x2x3^2 ehr_3_x2x3^2 h*_3_x2x3^2 ehr_4_x2x3^2 h*_4_x2x3^2 ehr_5_x2x3^2 h*_5_x2x3^2 ehr_6_x2x3^2 h*_6_x2x3^2
\(x_2^2x_3\) ehr_1_x2^2×3 h*_1_x2^2×3 ehr_2_x2^2×3 h*_2_x2^2×3 ehr_3_x2^2×3 h*_3_x2^2×3 ehr_4_x2^2×3 h*_4_x2^2×3 ehr_5_x2^2×3 h*_5_x2^2×3 ehr_6_x2^2×3 h*_6_x2^2×3
\(x_2^3\) ehr_1_x2^3 h*_1_x2^3 ehr_2_x2^3 h*_2_x2^3 ehr_3_x2^3 h*_3_x2^3 ehr_4_x2^3 h*_4_x2^3 ehr_5_x2^3 h*_5_x2^3 ehr_6_x2^3 h*_6_x2^3
\(x_3^3\) ehr_1_x3^3 h*_1_x3^3 ehr_2_x3^3 h*_2_x3^3 ehr_3_x3^3 h*_3_x3^3 ehr_4_x3^3 h*_4_x3^3 ehr_5_x3^3 h*_5_x3^3 ehr_6_x3^3 h*_6_x3^3