1,2

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 counts 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.

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)).

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

Sequence in context: A061639 A008995 A111236 this_sequence A006907 A052946 A026152

Adjacent sequences: A164358 A164359 A164360 this_sequence A164362 A164363 A164364

nonn

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