Hi,

Mathimagics!

Mathimagics wrote:The problematic transformations are those that involve row/col permutations within individual bands (as against ALL bands).

It's true that these may or may not yield another SudokuP.

But they can't result in automorphisms - this is true for all Sudoku's, not just these restricted forms, which is why the Russell & Jarvis tables show 0 for the number of invariant grids for all of these transformations.

I think that resolves that question!

Yes, such "non-universal" transformations cannot result to automorphisms. But automorphisms themselves are not the only obstacles, preventing us from counting essentially different grids.

Suppose, we have a set

G of transformations, acting on set

E of elements. Assume, that set

E is closed under set

G of transformations, i.e. result of any transformation of any element belongs to

E. Let's consider orbits of

E elements - results of all transformations applied to single element

e. If all orbits are disjoint sets and each orbit contains |

G| elements, the task of enumeration essentially different elements is simple. We need to divide |

E| by |

G| to get number of essentially different elements. But unfortunately in real life some elements can have automorphisms, decreasing size of orbits. Ignorance of automorphisms leads us to

underestimation of true number of essentially different elements.

Suppose, we have another set

H of transformations, acting on the same set

E of elements, such that set

H doesn't provide closure of

E. I.e. result of applying transformation

h, belonging to

H, to element

e can belong or not to set

E. Such "wrong" transformations can connect orbits of set

E elements under transformations from set

G. So, ignorance of "non-universal" transformations leads us to

overestimation of true number of essentially different elements. The "non-universal" transformations accounting problem is not related to automorphisms accounting problem.

Serg