Question $9)$ Stack of Boxes: A stable stack of boxes of height $n$ is a stack consisting of exactly

$n$ boxes of unit height stacked vertically in such a way that no bigger box is placed on a

smaller box. Note that if two boxes b$1$ and b$2$ have equal sizes, you can put b$1$ above b$2,$ or

below b$2,$ in the stable stack. You can see an example of a stable stack of boxes here:

$(20$ points$)$

Here two of the boxes $($the green boxes$)$ have equal sizes. The height of the stack is $5$ since

we have $5$ boxes. The height of the biggest box is $1$ and the height of the smallest box is $5.$

We have an infinite number of boxes of sizes $1,2,$dots,$m.$ Propose a dynamic programming

algorithm to calculate the number of different stable stacks of heightn that can be built from

these boxes, with a restriction that you can use at most k boxes of each size in the stack.