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.

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

Offset: 0

Author

Andrey Zabolotskiy, Sep 29 2017

Keywords

• nonn
• tabl
• hard
• more

Comments

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].

Examples

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

Links

• 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

Crossrefs

Cf. A006227, A293060, A293062, A293063.