1,2

Consider the set of 4 x 4 matrices with integer entries of a fixed determinant n. The group GL(4, \Z) acts on the right by multiplication. Similarly, the symmetric group S_4 acts on the left via multiplication by permutation matrices. The entry a_n is the number of elements in the double orbit space S_4\det^{-1}(n)/GL(4,\Z). The sequence a_n also counts the number of isomorphism classes of simplicial cones in \Z^4 of a certain index, or alternatively the number of affine toric varieties in dimension 4 arising from simplicial cones.

Atanas Atanasov, Table of n, a(n) for n=1..50

For n = 2, three representatives are [4,4]((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,1,2)), ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,1,1,2)) and ((1,0,0,0),(0,1,0,0),(0,0,1,0),(1,1,1,2)).

A162158 is the relevant sequence in dimension 3.

Sequence in context: A018852 A060092 A035283 this_sequence A004782 A116040 A036666

Adjacent sequences: A162156 A162157 A162158 this_sequence A162160 A162161 A162162

nonn

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