Expected Number of SCCs

Chef has a matrix $A$ with size $M \times N$. He wants to fill each of its cells with a random integer between $1$ and $K$ (inclusive); these integers are independent. Then, Chef builds a directed graph on this matrix: for each ordered pair of side-adjacent cells of the matrix such that the value in the first cell is greater than or equal to the value in the second cell, he adds an edge from th