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

1, 3, 7, 16, 18, 37, 36, 83, 85, 116, 101, 262, 154, 264, 337, 476, 305, 657, 409, 894, 788, 851, 682, 1778, 1037, 1338, 1530, 2123, 1288, 3006, 1550, 3083, 2622, 2799, 2969, 5403, 2544, 3821, 4155, 6591, 3399, 7441, 3891, 7172, 7652, 6552, 5012, 12605, 6512, 10047

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