A sweet corn sprinkler in square (I,J) will sprinkle all squares to the bottom-left: i.e. (i,j) withI≤i and j≤J.
An alfalfa sprinkler in square (I,J) will sprinkle all squares to the top-right: i.e.(i,j) withi≤I and J≤j.
A square sprinkled by one or multiple sweet corn sprinklers can grow sweet corn; a square sprinkled by one or multiple alfalfa sprinklers can grow alfalfa. But a square sprinkled by both types of sprinklers (or neither type) can grow nothing.
Help FJ determine the number of ways (modulo 109+7) to install sprinklers in his field, at most one per square, so that every square is fertile (i.e., sprinkled by exactly one type of sprinkler).
Some of the squares are already occupied by woolly cows; this doesn't prevent these squares from being fertile, but no sprinklers can be installed in such squares.
2 .. ..
28
Here are all fourteen possibilities when sweet corn can grow at (1,1)(1,1).
CC .C CA CC .C CA CA C. CA C. CC .C CC .C CC, CC, CC, .C, .C, .C, CA, CA, .A, .A, C., C., .., ..
4 ..W. ..WW WW.. ...W
2304
This satisfies the constraints for the first subtask described below.