Hi

Serg!

Serg wrote:The main problem of your method of essentially different SudokuP solution grids enumeration is your attempt to use "conjugacy classes" technique and Burnside's Lemma, but because basic requirements of Burnside's enumeration method are not met, it's not clear may we use conjugacy classes in this case.

I am not using Burnside's Lemma, we agree that can't be used here. But we are still basically counting automorphic grids, and the conjugacy classes allow us to reduce the problem because only a small number of the 3,359,232 transformation elements can possibly be automorphic.

Serg wrote: What are "26 S-classes" you are talking about in your post?

The 3,359,232 group elements for the Sudoku symmetry group partition into 275 conjugacy classes, and automorphism only occurs in 26 of these.

We use "S-class" to distinguish Sudoku group conjugacy classes from the "P-classes" of the "SudokuP" symmetry group (which has 2592 elements, and 54 conjugacy classes).

The class numbers we use correspond to the classes in the Russell & Jarvis paper quoted above, so these get used a lot in this forum.

So it is legitimate, and indeed necessary, to use conjugacy classes for counting problem based on S-equivalence, unless you want to waste time testing all 3,359,232 group elements.

For counting essentially different Sudoku grids, Burnside's Lemma reduces the problem still further, we need only count the automorphic grids in ONE element of each conjugacy class (ie one of the 26, all others we can ignore) and take the average over the group.

For counting essentially different SudokuP grids under S-equivalence, no we can't use the Burnside "trick", but we can still restrict ourselves to thos 26 classes, we just have to count automorphisms in every element of those classes. I call this "unfolding" the classes.