Really not sure why the state space would only grow as n^k / k!. As well as what n and k are in this case. Adding more tiles or more actors would both dramatically increase the size of the state space for that input.
That question should go to the parent comment. I only corrected the assumption.
On your comment though, I don't think there's much of "drama" in increasing the state space. Really it is just under 2 bits per cell by width by height. I would say it grows exponentially to the size of the board.
