Sweet Rectangles

Chef recently finished a birthday cake, which can be represented by a grid with $N$ rows (numbered $1$ through $N$) and $M$ columns (numbered $1$ through $M$). For each valid $i$, Chef placed a candle with height $a_{i,j}$ in the cell in row $i$ and column $j$. A rectangle with the top left corner in row $r_1$ and column $c_1$ and the bottom right corner in row $r_2$ and column $c_2$ ($1 \le r_