Bearing in mind

Serg's gentle (but absolutely factual) insistence regarding the true nature of Burnside's Lemma, I finally decided it was worth switching from Automorphism Enumeration to actual orbit counting.

Again,

Serg is correct when he stated that:

serg wrote:When we count non-equivalent in some sense or different in some sense puzzles/grids/patterns for given transformations group, we are counting orbits.

Using the only available enumeration of individual orbits (for the case of 20 clues, given by Gordon Foyle

here), I was able to determine that we can connect orbits if, and only if, for some image in the orbit:

- the 3x3 corner pattern has DD3 symmetry

- the 3x3 corner pattern does not have full symmetry, ie its horizontal reflection is different

So we need to distingish between non-reflecting cases (DD3 type 1), and fully-symmetric cases (DD3 type 2). The DD3 type 1 patterns are those box 1 patterns given above by

eleven above, the DD3 type 2 patterns are his box 2 patterns.

When I run the orbit counter with this orbit-joining tweak, I get results that correspond exactly with

JPF's table.

My results:

[/code]

Clues Patterns Orbits

----------------------

0 1 1

1 1 1

4 8 4

5 8 4

8 34 10

9 34 10

12 104 24

13 104 24

16 253 52

17 253 52

20 512 98

21 512 98

24 888 165

25 888 165

28 1344 246

29 1344 246

32 1794 323

33 1794 323

36 2128 380

37 2128 380

40 2252 402

41 2252 402

----------------------

32768 6016

[/code]

I guess what this shows is that we can arrive at the desired result without having to check orbits under

every VPT transformation, we only need to do it over FSPT, and check for explicit orbit connections (which the evidence suggests are possible only via "swap r1,r3" transformations on the applicable orbit images), so that's quite a saving.

I am thinking about how this approach might be applied to the SudokuP problem.

PS: the reason I say "only the 20-clues" case is available is that the link to Gordon Royle's full listing of S-equivalence classes no longer works, and I can't track it down. But I'm confident that they will all match up.