A293062 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, not counting enantiomorphs.

2, 5, 7, 31, 31, 80, 122, 360, 528, 1594, 1025

Offset: 0

Author

Andrey Zabolotskiy, Sep 29 2017

Keywords

• nonn
• tabl
• hard
• more

Comments

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. 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 (or G_{n+1,n,k}; the difference arises only when we consider enantiomorphism), except for the case k=n, when it is G_n^1 (or G_{n+1,n}).

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]. See Litvin for the cases when there are no enantiomorphs: rows 1-2, T(3,2). For T(3,1), see, e.g., [Palistrant & Jablan, 1991].

Examples

			The triangle begins:
     2;
     5,   7;
    31,  31,  80;
   122, 360, 528, 1594;
  1025, ...
		

Links

• H. Grimmer, Comments on tables of magnetic space groups, Acta Cryst., A65 (2009), 145-155.
• D. B. Litvin, Magnetic Group Tables
• A. F. Palistrant and S. V. Jablan, Enantiomorphism of three-dimensional space and line multiple antisymmetry groups, Publications de l'Institut Mathématique, 49(63) (1991), 51-60.
• B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
• Index entries for sequences related to groups

Crossrefs

Cf. A293060, A293061, A293063.

Formula

T(n,n) = A293060(n+1,n).