Distribution of objects in increasing order.

I know that we can distribute $$n$$ chocolates among $$k$$ children in $${n+k-1 \choose k-1}$$ ways which is the stars and bars problem. But what if the chocolates should be given in increasing order? I mean the 1st boy gets less chocolates than the second boy and so on.

If $$a_1+a_2+a_3+\cdots+a_k=n$$, then in how many ways can we distribute $$n$$ such that $$a_1?

If such a numerical problem is given, we can reduce $$n$$ to a smaller number and build various cases to solve it. But is there some generalized method to solve this kind of problems?

• Let $x_1=a_1, x_2=a_2-a_1, x_3 = a_3-a_2,\dots, x_k=a_k-a_{k-1}$. We equivalently have that $kx_1+(k-1)x_2+(k-2)x_3+\dots+2x_{k-1}+x_k=n$, but now the only restriction is $x_1\ge 0$ and $x_i>0$ for $i=2,\dots,k$. With a little more finagling, that reduces to this previously asked question: math.stackexchange.com/questions/2053319/…. – Mike Earnest May 10 '19 at 3:55
• $<$ (strictly increasing) or $\le$ (in order)? – Peter Taylor May 10 '19 at 12:23
• @PeterTaylor, strictly increasing. – tomriddle99 May 10 '19 at 14:33

The search term you need is partitions of $$n$$ into $$k$$ distinct parts".
The easy way to calculate them is a 3-variable recursion with $$n$$, $$k$$, and an upper bound on the largest part.