Combinatorial representation of function

Question

Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0.
The function can be defined in one of the following equivalent ways.

• The number $f(x,y,z)$.

• The number of all partitions (meaning: without considering the ordering) $x = a_1 + \ldots + a_y$, where $a_i \in \mathbb{Z}_{\geq 0}$ and $a_i \leq z$ for all $i$.

• The sum over the number of all partitions $x = a_1 + \ldots + a_l$, where $a_i \in \mathbb{Z}_{\geq 1}$ and $a_i \leq z$ for all $i$, where $l$ runs from $0$ to $k$.

• The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls.

Can we write $f$ as a function from $\mathbb Z^3 \to \mathbb Z$ ? probably recursively.? Or represent it somehow combinatorially using some partition function etc.?

PS: I am not looking for q-binomial formula because that gives " The number of ways of throwing $x$ distinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls." Instead, I am asking for "The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls. "

Answer 1

I'll rename all three of your variables; you are asking for the number of partitions of $k$ that fit into an $m \times n$ box. This is famously known to be the coefficient of $q^k$ in the $q$-binomial coefficient

$${m+n \choose m}_q = \frac{[m+n]_q!}{[m]_q! [n]_q!}$$

where $[n]_q! = \prod_{i=1}^n \left( \frac{q^i - 1}{q - 1} \right)$ is the $q$-factorial. Much is known about these, including a $q$-analog of Pascal's identity that leads to a recurrence.

Answer 2

You can obtain a recursion by conditioning on the largest part: $$f(x,y,z) = \sum_{p \le z} f(x-p,y-1,p)$$