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A173824 Number of four-dimensional simplical toric diagrams with hypervolume n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
PROG
(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)])
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.
Sequence in context: A200742 A178729 A124108 * A356603 A097211 A092945
KEYWORD
nonn
AUTHOR
Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Feb 25 2010
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
STATUS
approved

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Last modified May 7 05:04 EDT 2024. Contains 372300 sequences. (Running on oeis4.)