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A293062
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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.
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4
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2, 5, 7, 31, 31, 80, 122, 360, 528, 1594, 1025
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OFFSET
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0,1
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COMMENTS
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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].
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LINKS
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FORMULA
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EXAMPLE
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The triangle begins:
2;
5, 7;
31, 31, 80;
122, 360, 528, 1594;
1025, ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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