A164361 The number of elements in S_5\det^{-1}(n)/GL(5,\Z), where we take det : M_{5 \x 5}(\Z) \rightarrow \Z.

1, 4, 10, 29, 32, 89, 78, 236, 233, 381, 302, 1054, 518, 1109, 1377, 2350, 1274, 3649, 1872, 5280

Offset

1,2

Comments

Consider the set of 5 x 5 matrices with integer entries of a fixed determinant n. The group GL(5, \Z) acts on the right by multiplication. Similarly, the symmetric group S_5 acts on the left via multiplication by permutation matrices. The entry a_n is the number of elements in the double orbit space S_5\det^{-1}(n)/GL(5,\Z). The sequence a_n also gives the number of isomorphism classes of simplicial cones in \Z^5 of a certain index, or alternatively the number of affine toric varieties in dimension 5 arising from simplicial cones.

Links

Table of n, a(n) for n=1..20.

Example

For n = 2, four representatives are [5,5]((1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,1,2)), [5,5]((1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,1,1,2)), [5,5]((1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,1,1,1,2)), and [5,5]((1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(1,1,1,1,2)).

Crossrefs

A162158 and A162159 are the relative sequences in dimensions 3 and 4 respectively.

Sequence in context: A239577 A008995 A111236 * A006907 A305204 A327590

Adjacent sequences: A164358 A164359 A164360 * A164362 A164363 A164364

Keyword

nonn

Author

Atanas Atanasov (ava2102(AT)columbia.edu), Aug 14 2009

Status

approved