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A162158 The number of elements in S_3\det^{-1}(n)/GL(3,\Z), where we take det : M_{3 \x 3}(\Z) \rightarrow \Z. 3


%S 1,2,4,7,8,11,14,21,23,25,28,43,38,45,59,66,60,76,74,101,107,99,104,

%T 153,135,135,163,183,160,211,182,227,241,221,277,311,254,273,329,381,

%U 308,393,338,411,476,391,400,546,477,508,543,561,504,610,643,703,671

%N The number of elements in S_3\det^{-1}(n)/GL(3,\Z), where we take det : M_{3 \x 3}(\Z) \rightarrow \Z.

%C 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.

%H Atanas Atanasov, <a href="/A162158/b162158.txt">Table of n, a(n) for n=1..210</a>

%e 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)).

%Y Cf. A162159. - Atanas Atanasov (ava2102(AT)columbia.edu), Jun 29 2009

%K nonn

%O 1,2

%A Atanas Atanasov (ava2102(AT)columbia.edu), Jun 26 2009

%E Terms a(24) and beyond from b-file by _Andrew Howroyd_, Feb 05 2018

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Last modified October 18 03:25 EDT 2021. Contains 348065 sequences. (Running on oeis4.)