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A162158
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The number of elements in S_3\det^{-1}(n)/GL(3,\Z), where we take det : M_{3 \x 3}(\Z) \rightarrow \Z.
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3
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1, 2, 4, 7, 8, 11, 14, 21, 23, 25, 28, 43, 38, 45, 59, 66, 60, 76, 74, 101, 107, 99, 104, 153, 135, 135, 163, 183, 160, 211, 182, 227, 241, 221, 277, 311, 254, 273, 329, 381, 308, 393, 338, 411, 476, 391, 400, 546, 477, 508, 543, 561, 504, 610, 643, 703, 671
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OFFSET
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1,2
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COMMENTS
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Consider the set of 3 x 3 matrices with integer entries of a fixed determinant n. The group GL(3, \Z) acts on the right by multiplication. Similarly, the symmetric group S_3 acts on the left via multiplication by permutation matrices. The entry a_n is the number of elements in the double orbit space S_3\det^{-1}(n)/GL(3,\Z). The sequence a_n also gives the number of isomorphism classes of simplicial cones in \Z^3 of a certain index, or alternatively the number of affine toric varieties in dimension 3 arising from simplicial cones.
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LINKS
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EXAMPLE
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For n = 2, two orbit representatives are ((1,0,0),(0,1,0),(0,1,2)) and ((1,0,0),(0,1,0),(1,1,2)). For n = 3, we have ((1,0,0),(0,1,0),(0,1,3)), ((1,0,0),(0,1,0),(0,2,3)), ((1,0,0),(0,1,0),(1,1,3)) and ((1,0,0),(0,1,0),(2,2,3)).
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CROSSREFS
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Cf. A162159. - Atanas Atanasov (ava2102(AT)columbia.edu), Jun 29 2009
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KEYWORD
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nonn
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AUTHOR
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Atanas Atanasov (ava2102(AT)columbia.edu), Jun 26 2009
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EXTENSIONS
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STATUS
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approved
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