I know how to find the number of solutions to the equation:
$a_1 + a_2 + \dots + a_k = n$
where $n$ is a given positive integer and $a_1$, $a_2$, $\dots$, $a_n$ are positive integers. The number of solutions to this equation is:
$\binom{n - 1}{k - 1}$
This can be imagined as $n$ balls arranged on a straight line and selecting $k - 1$ gaps from a total of $n - 1$ gaps between them as partition boundaries. The $k - 1$ partition boundaries divide the $n$ balls into $k$ partitions. The number of balls in the $i$th partition is $a_i$.
Now, I don't know how to find the number of solutions to the same equation when we have an additional constraint: $0 < a_1 \leq a_2 \leq \dots \leq a_k.$ Could you please help me?