Tom's partials combinations

This page is a copy-paste (with the table re-formatted) of Tom's message #7108 on eternity_two yahoo group.
It's about 2x2, 3x3, 4x4 , and 5x5 (yes !) combinations count :

I have extracted some data from my table of partials for the puzzle and reproduced it below.

There are four pairs of columns for each type of partial:

1. If no tiles are fixed
2. If the fixed piece (#139) is locked in place
3. If the fixed piece plus the first two hints are locked down
4. If the fixed piece and all four hints are locked down

The two numbers in each pair are:

1. The number of partials calculated by my solver that doesn't do any extra domain reduction. Many of these numbers have been confirmed in this forum by other members (e.g. eternity2blogger, okifinoki, Michel Gaillard and others)
2. The second number in each pair is the number of partials that result when my solver does some domain reduction first. This domain reduction is a combination of two effects: a partial ARC consistency followed by targeted 3x3 solvers executed about any location that has a low domain count.

Here is the table:

You will see that the domain reduction only effects the partials that include (or are adjacent to) one of the fixed pieces.

Thanks to Tom for this nice work.