A173878 -id:A173878 - cp's OEIS frontend

A159842 Number of symmetrically-distinct supercells (sublattices) of the fcc and bcc lattices (n is the "volume factor" of the supercell).

1, 2, 3, 7, 5, 10, 7, 20, 14, 18, 11, 41, 15, 28, 31, 58, 21, 60, 25, 77, 49, 54, 33, 144, 50, 72, 75, 123, 49, 158, 55, 177, 97, 112, 99, 268, 75, 136, 129, 286, 89, 268, 97, 249, 218, 190, 113, 496, 146, 280, 203, 333, 141, 421, 207, 476, 247, 290, 171, 735

Offset: 1

Gus Hart (gus_hart(AT)byu.edu), Apr 23 2009

nonn

The number of fcc/bcc supercells (sublattices) as a function of n (volume factor) is equivalent to the sequence A001001. But many of these sublattices are symmetrically equivalent. The current sequence lists those that are symmetrically distinct.
Is this the same as A045790? - R. J. Mathar, Apr 28 2009
This sequence also gives number of sublattices of index n for the diamond structure - see Hanany, Orlando & Reffert, sec. 6.3 (they call it the tetrahedral lattice). Indeed: the diamond structure consists of two interpenetrating fcc lattices, and all sites of any sublattice should belong to the same fcc lattice because every sublattice is inversion-symmetric. - Andrey Zabolotskiy, Mar 18 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
J. Davey, A. Hanany and R. K. Seong, Counting Orbifolds, J. High Energ. Phys., 2010, 10; arXiv:1002.3609 [hep-th], 2010.
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, J. High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th], 2010.
Gus L. W. Hart and Rodney W. Forcade, Algorithm for generating derivative structures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115.
Materials Simulation Group at Brigham Young University, Derivative structure enumeration library.
Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda and Isao Tanaka, Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram, J. Chem. Phys. 153, 104109 (2020); preprint: Derivative structure enumeration using binary decision diagram, arXiv:2002.12603 [physics.comp-ph], 2020.
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to f.c.c. lattice
Index entries for sequences related to b.c.c. lattice

Cf. A045790.
Cf. A001001.
Cf. A003051, A145393, A145391, A145398, A300782, A300783, A300784.
Cf. A173824, A173877, A173878.

def dc(f, *r): # Dirichlet convolution of multiple sequences
    if not r:
        return f
    return lambda n: sum(f(d)*dc(*r)(n//d) for d in range(1, n+1) if n%d == 0)
def fin(*a): # finite sequence
    return lambda n: 0 if n > len(a) else a[n-1]
def per(*a): # periodic sequences
    return lambda n: a[n%len(a)]
u, N, N2 = lambda n: 1, lambda n: n, lambda n: n**2
def a(n): # Hanany, Orlando & Reffert, sec. 6.3
    return (dc(u, N, N2)(n) + 9*dc(fin(1, -1, 0, 4), u, u, N)(n)
            + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)
            + 6*dc(fin(1, -1, 0, 2), u, u, per(0, 1, 0, -1))(n))//24
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Mar 18 2018

Terms a(20) and beyond from Andrey Zabolotskiy, Mar 18 2018

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

Original entry on oeis.org

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

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

nonn

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

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.

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.

# 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)])

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

A173877 Number of five-dimensional simplical toric diagrams with hypervolume n.

Original entry on oeis.org

1, 3, 6, 17, 13, 40, 27, 106, 78, 127, 79, 391, 129, 321, 358, 832, 285, 1070, 409, 1549

Offset: 1

Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Mar 01 2010

Also gives the number of distinct abelian orbifolds of C^6/Gamma, Gamma in SU(6).

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.
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.

Cf. A003051 (No. of two-dimensional triangular toric diagrams of area n), A045790 (No. of three-dimensional tetrahedral toric diagrams of volume n), A173824 (No. of four-dimensional simplical toric diagrams of hypervolume n), A173878.

a(9) corrected, a(15)-a(20) added from Hanany & Seong 2011 by Andrey Zabolotskiy, Jun 30 2019

cp's OEIS Frontend

A159842 Number of symmetrically-distinct supercells (sublattices) of the fcc and bcc lattices (n is the "volume factor" of the supercell).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage

Extensions

A173877 Number of five-dimensional simplical toric diagrams with hypervolume n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions