Number of combinations to draw the sum $n$ from set of $\{1,\dots,n\}$

Question

How many different combinations $m$ are there to draw numbers $n_i$ with $n=\sum^{N}_i n_i$ from the set $\{1,\dots,n\}$ without regarding order and without replacement, and for a given number of draws $N$.

I am studying Euler's pentagonal number theorem and try to think about alternative ways to understand the multiplicities of the powers in the Euler function.

I have been drawing some diagrams for $n = 11$ and $N = 1, 2, 3, 4$ but while seeing some graphical patterns it seems very hard to derive numbers from these.

Answer 1

As @quasi's answer implies, the question can be obviously reformulated to: "What is the number $m$ of partitions of $n$ into $N$ distinct parts?".

In that form for $N = 2, 3$ partial sequences are contained e.g. in the OEIS sequences for $N=2$ and $N$=3 and, the latter also contains a concise formulation in terms of a generating function:

For each $N$ the sequences are formed by the coefficients of the terms $t^N$ in the product $$ \prod_{i=1}^{\infty}(1+t q^i)$$

which is identical to $(-t;q)_\infty$ in q-Pochhammer symbol notation (in Mathematica code you can get partial sequences up to size $N$ with CoefficientList[Coefficient[Product[1+t*x^i, {i,N}], t^N], x], by setting the appropriate value of $N$, sequence members larger than $N$ are incorrect for that purpose due to the truncation of the generating product.)

As @quasi pointed out a closed form for $N=2$, $s_N = 0,0,1,1,2,2,3,3,\dots = ({\lfloor}\frac{N-1}{2}{\rfloor})_N $ is obvious. For $N>2$ no closed form seems to be known. In Analogy to the case for the closely related partition number of n it can be conjectured that at least a recursive expression and maybe a convergent approximation exists.

Answer 2

For positive integers $n,k$ with $k \le n$, let $f(n,k)$ be the number of partitions of $n$ into $k$ distinct parts.

Then $f(n,k) = g(n,k,1)$, where $g(n,k,m)$ is the number of partitions of $n$ into $k$ distinct parts such that the smallest part is greater than or equal to $m$.

The values of $g$ can be computed recursively.

Here's a Maple implementation . . .