Partition problem where partition are in increasing order.

Question

For given $n$ and $S$, how many possible combinations are there such that:

$x_1 + x_2 + .. + x_n = S $ $\forall i, x_i \leq x_{i+1}$ $\&$ $x_i \geq 1$

For example, if $n$ = 3 and $S$ = 5, there are 2 possible combination:

• 1 + 1 + 3
• 1 + 2 + 2