6. (10 points) Color 7 cells of a $4 \times 4$ grid with the same color. If a column has colored cells, the cells must be colored from the bottom cell upwards. In adjacent columns, the number of colored cells in the left column must be greater than or equal to the number of colored cells in the right column (as shown in the figure). There are $\qquad$ ways to color the grid.
【Analysis】According to the requirements, 7 squares in a $4 \times 4$ grid are to be colored, with the number of colored squares on the left side being greater than or equal to the number on the right side. Break down 7 into the sum of several numbers, with the largest number on the left being 4, for example, $4+3=7$, coloring from the first column to the third column, there are 3 patterns;
$3+2+2=7$, coloring from columns $1$ and $2$, there are 2 patterns;
$3+2+1+1$, only coloring from the first column, there is 1 pattern;
$4+1+1+1$, only coloring from the first column, there is 1 pattern;
$4+2+1=7$, coloring from columns $1$ and $2$, there are 2 patterns;
Adding them up, we get the solution.
【Solution】Solution: As shown in the figure,
$3+2+1+1+2=9$ (patterns),
Answer: There are 9 patterns in total.
Therefore, the answer is: 9.