co.combinatorics - unique integer partitions

For fixed $k$ and large $n$ this is pretty doable. You want to find solutions to

$$x_1 + x_2 + ... + x_k = n$$

where $x_1 ge x_2 ge ... ge x_k$. Letting $y_i = x_i - x_{i+1}$ and $y_k = x_k$, this is equivalent to finding solutions to

$$y_1 + 2y_2 + ... + ky_k = n$$

where $y_i ge 0$. If $p_k(n)$ denotes the number of ways to do this, it follows by a standard generating function trick that

$$sum_{k ge 0} p_k(n) x^n = frac{1}{(1 - x)(1 - x^2)...(1 - x^k)}.$$

In principle one can find the partial fraction decomposition of the RHS, allowing us to write $p_k(n)$ as a quasi-polynomial.