1,2

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 counts 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.

Atanas Atanasov, Table of n, a(n) for n=1..210

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)).

A162159 [From Atanas Atanasov (ava2102(AT)columbia.edu), Jun 29 2009]

Sequence in context: A074204 A030355 A080704 this_sequence A018552 A030773 A118248

Adjacent sequences: A162155 A162156 A162157 this_sequence A162159 A162160 A162161

nonn

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