Number of non negative integer solutions of $l_1+l_2+...+l_m=n$ for constraint $l_i\leq k$

Question

How do I find a closed form solution to number of non negative integer solutions of $l_1+l_2+...+l_m=n$ for constraint $l_i\leq k$. I know that without the constraint, the answer is $$n+m-1 \choose{m-1}$$. However, How would I deal with the constraint here?

Answer 1

Although not exactly a closed solution, stars and bars can be extended with inclusion-exclusion.

This will lead to the following formula:

$$ \sum_{j=0}^m(-1)^j\binom mj\binom{n-j(k+1)+m-1}{m-1}\; $$

$j$ counts the number of variables where the constraint is violated, and you are to continue only while the upper binomial index $\ge$ lower on

If you don't like "continue only while the upper binomial index $\ge$ lower one.."

define $J = \lfloor \frac{m}{k}\rfloor$ to make the formula

$$ \sum_{j=0}^J(-1)^j\binom mj\binom{n-j(k+1)+m-1}{m-1}\; $$

or put everything into one formula, thus

$$ \sum_{j=0}^{\lfloor\frac{m}{k}\rfloor}(-1)^j\binom mj\binom{n-j(k+1)+m-1}{m-1}\; $$