A293062 -id:A293062 - cp's OEIS frontend

A293060 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, not counting enantiomorphs.

1, 2, 2, 10, 7, 17, 32, 67, 80, 219, 227, 343, 1076, 1594, 4783, 955

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

T(n,0) count n-dimensional crystallographic point groups (i.e., left border is A004028), T(n,n) count n-dimensional space groups (i.e., right border is A004029). The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., symmetry groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional space groups.
The Bohm symbols for these groups are G_{n,k}, except for the case k=n, when it is G_n.
Some groups have their own names:
T(2,1): frieze groups
T(2,2): wallpaper groups
T(3,1): rod groups
T(3,2): layer groups
See [Palistrant, 2012, p. 476] for row 4.

			The triangle begins:
    1;
    2,   2;
   10,   7,   17;
   32,  67,   80,  219;
  227, 343, 1076, 1594, 4783;
  955, ...

M. I. Aroyo et al, Bilbao Crystallographic Server
International Union of Crystallography, International Tables for Crystallography, volumes A and E.
A. F. Palistrant, Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57 (2012), 471-477.
W. Plesken and T. Schulz, CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
Wikipedia: Space group, Crystallographic point group, Line group, Frieze group, Wallpaper group, Rod group, Layer group
Index entries for sequences related to groups

Cf. A004028, A004029, A293061, A293062, A293063.

A293061 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, counting enantiomorphs.

Original entry on oeis.org

1, 2, 2, 10, 7, 17, 32, 75, 80, 230, 271, 343, 1091, 1594, 4894, 955

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

T(n,0) count n-dimensional crystallographic point groups, T(n,n) count n-dimensional space groups (i.e., right border is A006227). The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., symmetry groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional space groups.
The Bohm symbols for these groups are G_{n,k}, except for the case k=n, when it is G_n.
Some groups have their own names:
T(2,1): frieze groups
T(2,2): wallpaper groups
T(3,1): rod groups
T(3,2): layer groups
[Palistrant, 2012, p. 476] gives correct T(4,k), k=0,1,2,3 but incorrect T(4,4). For correct value of T(4,4), see [Souvignier, 2006, p. 80].

			The triangle begins:
    1;
    2,   2;
   10,   7,   17;
   32,  75,   80,  230;
  271, 343, 1091, 1594, 4894;
  955, ...

M. I. Aroyo et al, Bilbao Crystallographic Server
International Union of Crystallography, International Tables for Crystallography, volumes A and E.
A. F. Palistrant, Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57 (2012), 471-477.
W. Plesken and T. Schulz, CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
Wikipedia: Space group, Crystallographic point group, Line group, Frieze group, Wallpaper group, Rod group, Layer group
Index entries for sequences related to groups

Cf. A006227, A293060, A293062, A293063.

A293063 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, counting enantiomorphs.

Original entry on oeis.org

2, 5, 7, 31, 31, 80, 122, 394, 528, 1651, 1202

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

Magnetic groups are also known as antisymmetry groups, or black-white, or two-color crystallographic groups.
T(n,0) count n-dimensional magnetic crystallographic point groups, T(n,n) count n-dimensional magnetic space groups (A307291). The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., magnetic groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional magnetic space groups.
The Bohm-Koptsik symbols for these groups are G_{n,k}^1, except for the case k=n, when it is G_n^1.
T(2,1) are band groups.
T(3,3) are Shubnikov groups.
For T(n,0) and T(n,n), see [Souvignier, 2006, table 1]. For rows 1-3, see Litvin.

			The triangle begins:
     2;
     5,   7;
    31,  31,  80;
   122, 394, 528, 1651;
  1202, ...

H. Grimmer, Comments on tables of magnetic space groups, Acta Cryst., A65 (2009), 145-155.
D. B. Litvin, Magnetic Group Tables
B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
Index entries for sequences related to groups

Cf. A293060, A293061, A293062, A307291.

A307292 Number of black-and-white colored (or proper magnetic) space groups in dimension n, not counting enantiomorphs.

Original entry on oeis.org

0, 3, 46, 1156, 51987

Offset: 0

N. J. A. Sloane, Apr 08 2019

R. L. E. Schwarzenberger, Colour symmetry, Bulletin of the London Mathematical Society 16.3 (1984): 216-229.
B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.

For the case when enantiomorphs are counted see A307290.

Cf. A004029, A293062.

a(n) = A293062(n, n) - 2 * A004029(n). - Andrey Zabolotskiy, Apr 09 2019

a(0), a(4) from Andrey Zabolotskiy, Apr 09 2019

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A293060 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, not counting enantiomorphs.

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A293061 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, counting enantiomorphs.

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A293063 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, counting enantiomorphs.

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Author

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Comments

Examples

Links

Crossrefs

A307292 Number of black-and-white colored (or proper magnetic) space groups in dimension n, not counting enantiomorphs.

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