Let $P(n,k,m,z)$ be the number of partitions, that a positive integer $n$ can be partitioned into when each partition has exactly $k$ parts and when all of these parts are $\leq z$ and $\ge m$.
For example: $P(10,3,1,4) = 2$,
...because only two such triplets are possible: {4,4,2}, {4,3,3}.
Q) Is there a closed form or an asymptotic solution for $P(n,k,m,z)$, when $ k \leq n \leq kz$ and $ n \geq km$ ?
EDIT:
The answer in that post presents a beautiful idea to calculate the number of unrestricted integer partitions. It is based on:
- The generating function: $\sum_{n=0}^{\infty}p\left(n\right)x^{n}=\prod_{n=1}^{\infty}\frac{1}{\left(1-x^{n}\right)}$
combined with the: - Euler's Pentagonal theorem: $\prod_{n=1}^{\infty}\left(1-x^{n}\right)=\sum_{k=-\infty}^{\infty}\left(-1\right)^{k}x^{k\left(3k-1\right)/2}$
How can it be modified to calculate the number of integer partitions with the triple restriction by $k,m,z$ ?
P.S.
I have seen similar questions about the number of integer partitions with 1 restriction, e.g. number of parts in partition OR a restriction on values of the parts ...but never BOTH.
Please consider that before marking this question as a duplicate.