A162159 The number of elements in S_4\det^{-1}(n)/GL(4,\Z), where we take det : M_{4 \x 4}(\Z) \rightarrow \Z.

1, 3, 7, 16, 18, 37, 36, 83, 85, 116, 101, 262, 154, 264, 337, 476, 305, 657, 409, 894, 788, 851, 682

Offset

1,2

Comments

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 gives 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.

Links

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

Example

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)).

Crossrefs

A162158 is the relevant sequence in dimension 3.

Sequence in context: A060092 A035283 A184863 * A190890 A116040 A218276

Adjacent sequences: A162156 A162157 A162158 * A162160 A162161 A162162

Keyword

nonn

Author

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

Status

approved