Ordered stars and bars Find the number of ordered $8$-tuples of nonnegative integers $x_0 < x_1 < x_2 < \cdots < x_7$ such that $\sum_{i=0}^{7} x_i = 99$
The above question clearly cannot be answered with the classic stars and bars, and the substitution $y_i = x_i - i$ doesn't seem to help either. I cannot see how to progress. 
 A: Brain malfunction. You are looking for a partition into different parts with 7 parts of the integer 99
A: We can get it by generating function for partitions (as suggested by vonbrand), which is given by
$$
G(x)=\frac{x^{28}}{{\left(1-x\right)} {\left(1-x^{2}\right)} {\left(1-x^{3}\right)}{\left(1-x^{4}\right)}{\left(1-x^{5}\right)}{\left(1-x^{6}\right)}{\left(1-x^{7}\right)} {\left(1-x^{8}\right)}}
$$
and the required answer is $$[x^{99}]G(x)=207945$$
A: Let $F(m,k,b)$ be the number of strictly ordered $k$-tuples $0 \leq x_1 < \ldots < x_k \leq b$ with $\sum_{i=1}^k x_k = m$.
For $b < k-1$ these requirements are contradictory. The minimal sum of a valid $k$-tuple is $0 + 1 + \ldots + (k-1)$, which is $\frac{k(k-1)}{2}$. The maximal sum of a valid $k$-tuple is $b + (b-1) + \ldots + (b-k+1)$, which is $\frac{b(b+1) - (b-k)(b-k+1)}{2}$.
For $m = \frac{k(k-1)}{2}$ and $b \geq k-1$, there is exactly the one $k$-tuple $(0,1,\ldots,k-1)$.
It follows that $$\begin{eqnarray}
  (1)\quad& F(m,k,b) &=& 0 \quad \text{if } b < k-1 \text{ or } m < \tfrac{k(k-1)}{2} \text{ or } m > \tfrac{b(b+1) - (b-k)(b-k+1)}{2} \\
  (2)\quad& F(m,k,b) &=& 1 \quad \text{if } b \geq m \text{ and } k=1 \\
  (3)\quad& F(m,k,b) &=& 1 \quad \text{if } b \geq k-1 \text{ and } m = \tfrac{k(k-1)}{2} \text{.}
\end{eqnarray}
$$
Picking a fixed $x_k \in [0,b]$ means that $x_1,\ldots,x_{k-1}$ have to be less than $x_k$, i.e. bounded by $b-1$, and that their sum has to be $m-x_k$, which yields
 $$
  F(m,k,b) = \sum_{i=0}^b F(m - i, k-1, i-1) \text{.}
$$
(The sum's range for $i$ can be reduced further, but using the conditions under which $F(m-i,k-1,i-1)$ is zero)
Now one would have to see if this recursion can be brought into an explicit form.
