A173824 Number of four-dimensional simplical toric diagrams with hypervolume n.

1, 2, 4, 10, 8, 19, 13, 45, 33, 47, 30, 129, 43, 96, 108, 226, 78, 264, 102, 357, 226, 277, 163, 813, 260, 425, 436, 780, 297, 1092, 355, 1281, 678, 856, 712, 2215, 569, 1155, 1050, 2537, 752, 2544, 856, 2447, 2048, 1944, 1093, 5388, 1447, 3083, 2150, 3827

Offset: 1

Author

Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Feb 25 2010

Keywords

• nonn

Comments

Also gives the number of distinct abelian orbifolds of C^5/Gamma, Gamma in SU(5).

Links

• J. Davey, A. Hanany and R. K. Seong, Counting Orbifolds, J. High Energ. Phys. (2010) 2010: 10; arXiv:1002.3609 [hep-th], 2010.
• A. Hanany and R. K. Seong, Symmetries of abelian orbifolds, J. High Energ. Phys. (2011) 2011: 27; arXiv:1009.3017 [hep-th], 2010-2011. Table 5 gives a(1)-a(80), but the terms a(36) and a(65) there are apparently erroneous.
• Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.

Crossrefs

Cf. A003051 (No. of two-dimensional triangular toric diagrams of area n), A045790 (No. of three-dimensional tetrahedral toric diagrams of volume n), A173877, A173878.

Programs

Sage

# see Python in A159842 for the definition of dc, fin, per, u, N, N2
def fin_d(d):
    return fin(*(d.get(n+1, 0) for n in range(max(d))))
def a(n): # see Hanany & Seong 2011, Table 1 row D=5 and Table 9
    return (dc(u, N, N2, lambda n: n**3)(n) +
        10 * dc(u, u, N, N2, fin(1, -1, 0, 8))(n) +
        15 * dc(u, u, N, N, fin_d({1: 1, 2: -3, 4: 14, 8: -12, 16: 16}))(n) +
        20 * dc(u, u, N, per(0, 1, -1), fin(1, 0, -1, 0, 0, 0, 0, 0, 9))(n) +
        20 * dc(u, u, u, per(0, 1, -1), fin(1, -1, 0, 2), fin(1, 0, -1, 0, 0, 0, 0, 0, 3))(n) +
        30 * dc(u, u, u, per(0, 1, 0, -1), fin_d({1: 1, 2: -2, 4: 3, 16: 6, 32: -8, 64: 8}))(n) +
        24 * dc(u, per(0, 1, -1, -1, 1), per(0, 1, I, -I, -1), per(0, 1, -I, I, -1))(n)) / 120
print([a(n) for n in range(1, 100)])

Extensions

a(16) corrected, terms a(31) and beyond added from Hanany & Seong 2011 by Andrey Zabolotskiy, Jun 30 2019

a(36) corrected from 2202 to 2215 by Andrey Zabolotskiy, Sep 20 2022