Go is a classical Asian game, where the players take turn placing a black or white stone on a 19 by 19 grid of lines. How many different board configurations are there?

Given that:

The first intersection can be vacant.
The first intersection can contain a black stone.
The first intersection can contain a white stone.

Considering two intersections we have still three possibilities for each, giving us a total of nine.

We know the board is 19 by 19, so we can raise the number of possibilities to the number of intersections, in the case of the 19 by 19 board, we get: $3^{(19 \cdot 19)}$

Which is:
17408965065903192790718823807056436794660272495026354119482811870680105167618464984116279288988714938612096988816320780613754987181355093129514803369660572893075468180597603

Which is correct according to: https://en.wikipedia.org/wiki/Go_and_mathematics#Legal_positions

The original question comes from an interview (and is already answer by me with my reasoning), the question below is what I am curious in:
What I am curious to understand is how to calculate the legal positions, and why is it that as the board gets larger, the percentage of the positions that are legal decreases.

Question

Go is a classical Asian game, where the players take turn placing a black or white stone on a 19 by 19 grid of lines. How many different board configurations are there?

Given that:

The first intersection can be vacant. 
The first intersection can contain a black stone. 
The first intersection can contain a white stone.

Considering two intersections we have still three possibilities for each, giving us a total of nine.

We know the board is 19 by 19, so we can raise the number of possibilities to the number of intersections, in the case of the 19 by 19 board, we get: $3^{(19 \cdot 19)}$

Which is: 
17408965065903192790718823807056436794660272495026354119482811870680105167618464984116279288988714938612096988816320780613754987181355093129514803369660572893075468180597603

Which is correct according to: https://en.wikipedia.org/wiki/Go_and_mathematics#Legal_positions

The original question comes from an interview (and is already answer by me with my reasoning), the question below is what I am curious in:
What I am curious to understand is how to calculate the legal positions, and why is it that as the board gets larger, the percentage of the positions that are legal decreases.

kainui · Answer

My guess (without much thought so could be totally wrong) is that since when you play the game you are essentially able to make completely disconnected islands. But as the game progresses, these eventually expand outwards and 'collide'. So the larger the board, the more you're able to create these sort of collisions which are illegal in an actual game in more and more ways, while the more restricted the board is, the more forced in to making legal moves earlier happens. Imagine a 1x1 grid (haha) you can't make any illegal moves because there's no space to. Illegal moves are what comes out of having freedom to detach the board apart I think.

inkyvoyd · Answer

SPEAKING OF GO: https://en.wikipedia.org/wiki/AlphaGo

opcode · Answer

To get the Nature article (that is in the references of the Wikipedia link dropped by Inky) if anyone is interested: https://storage.googleapis.com/deepmind-data/assets/papers/deepmind-mastering-go.pdf @Kainui you have lost me, when you say collide, do you mean cut of a stone in which the opposite stone has no liberties? (Such as a mass group of black/white stones?) I went ahead and cheated a bit, and looked at the OEIS https://oeis.org/A094777, which still leaves me clueless...