Polyominoes 1

A polyomino is a geometric figure obtained by joining equal squares edge by edge.
We consider that one side of each square is painted: when comparing two polyominoes for symetry, the painted sides must always stay up.
Two polyominoes are equal if they differ only by a rotation of a multiple of $\frac{\pi }{2}$

To sort the polyominoes, we define the "canonical value" of a polyomino as follow:

• A polyomino is represented by a matrix of 0 and 1
• We consider all 4 rotations and keep those where the matrix is such as the number of rows is $\le$ the number of columns.
• We add a column of 0 at the end, flatten the values and consider them as a binary representation of some integer. The canonical value is the smallest of these integers.

For example, consider the polyomino
It can be represented either by $\left(\begin{array}{}0& 0& 1\\ 1& 1& 1\end{array}\right)$ or by $\left(\begin{array}{}1& 1& 1\\ 1& 0& 0\end{array}\right)$

The first representation yields the values $\{0,0,1,0,1,1,1,0\}$ = 46
The second representation yields the values $\{1,1,1,0,1,0,0,0\}$ = 232
Thus, the canonical value for this polyomino is 46.

With 4 squares, there are 7 possible polyominoes (sorted by their canonical values):

• 30
• 46
• 54
• 78
• 108
• 142
• 198

How many polyominoes with 8 squares are there?
What is the canonical value of the 666-th poylyomino?

Format answer: Count,Value

[My timing: 40 sec]