"N. J. A. Sloane" <njas@research.att.com> wrote: [...] :allowing (i quote) : : Relabelling entries; : Reflection; : Rotation; : Permutation of blocks of columns 1-3, 4-6 and 7-9; : Permutation of blocks of rows 1-3, 4-6 and 7-9; : Permutation of columns 1-3; : Permutation of rows 1-3; : Permutation of columns 4-6; : Permutation of rows 4-6; : Permutation of columns 7-9; : Permutation of rows 7-9. : :So there's a new sequence here: : :0 1 2 3 :1 1 x 5472730538 : :where x is some small number! My question is, what is x? :Using analogues of the above transformations, how many :inequivalent 4 X 4 grids are there? : :Hugo van der Sanden sent me an upper bound x <= 6, since any 4X4 :grid is equivalent to one of: Let me label them: A B C D E F :1234 1234 1234 1234 1234 1234 :3412 3412 3421 4312 4321 4321 :2143 2341 2143 2143 2143 2413 :4321 4123 4312 3421 3412 3142 : :but this can surely be reduced. So is x 1, 2 or 3? x is 2. I don't visualise these well, but I had time to write some code which found: A => permute columns (1-2), relabel (1-2) => E B => mirror antidiagonal, relabel (1-2-3) => C B => rotate -90', relabel (1-3-4) => D B => permute columns (1-2), permute rows (3-4), relabel (1-2) => F So {A, E} are identical, and {B, C, D, F} are identical. There is no way to transform A to B: the pattern of opposing pairs in A (12/21 and 34/43 vertically; 13/31 and 24/42 horizontally) is matched in only one axis in B (13/31 and 24/42 horizontally), and there is no transformation that can change that. So A and B are truly distinguishable, and a(2) = 2. Hugo