Thanks Ed/Frazer It had dawned on me (literally when waking up), that the problem was the 2 different types of 'equivalence' (as Ed stated)

I had been working from the 'classic' definition of symmetry equivalence: that members of an equivalence class can be permuted into each other by the allowed symmetries. (For Sudoku, the 9! * 2 * 6^8 symmetries.) Hence my problem in understanding the minicol permutation approach.

The dawning revealed that there is also a second type of equivalence:

Ed wrote:..the only way it can affect completion of the grid is through column constraints.

If you group the top band configurations by the way they can affect the completion of the rest of the grid, you also get equivalence classes: Each member of the equiv.class will have the same (counting) set of completions because all contribute the same constraints to the lower bands.

By focusing on the band and not the direct Sudoku One-Rule constraints, we achieve an equivalent problem, but much easier to work with.

We can apply the same principle to the first stack (blks 147). The same gang of 44 will apply, just transposed and with a minor renumbering. Counting all the solutions is reduced to the 44^2 combinations of completing blocks 5689.

Frazer wrote:...Ed's claim that counting-equivalence implies permutation-equivalence probably has no explanation

Ed's explanation is very close to a proof - given the appropriate defintiion of equiv.class, namely counting-equivalent, which refocuses the problem on the constraints propagated outside the band. Of course to find 'real' (generating) members of the counting-equivalent classes you still need to enumerate through the Sudoku compatible permutations of the (top) band.

If I restate (Ed's) definition of the two type of equivalence slightly:

1. Two examples of band 1 are permutation-equivalent if you can get from one to the other by, (Larry: omit: reordering minicol contents), reordering minicols within boxes, reordering boxes and/or relabelling digits.

2. Two examples of band 1 are counting-equivalent if they give rise to the same number of completions of the grid, (Larry: add i.e. they can be reduced to a canonical representative by the permutation reorderings + reordering minicol contents.)

then the following is clear:

Ed wrote:permutation equivalence implies counting equivalence. It just so happens that the converse is also true. . that [the] set of ops in [counting equivalence] should succeed in capturing all aspects of permutation equivalence,

Because we populate the counting-equivalence classes with

all the permutation-equivalence members (deduced from the 9!*72*32688) they (the counting-equivalence) classes must capture

all the permutation based combinations.

I think it is this pairing of the two equivalence types which provides the power to the T-Class band generation approach. We simplify the problem by using minicol content permutations and then at the appropriate points choose to use only Sudoku compatible members (gang of 44) when enumerating the solutions.