Let $P(z;k,\ell)$ denote the ordinary generating function of the partitions $\mathcal P$ with at most $k$ parts each at most $\ell$.
Hint. By distinguishing the cases $i_\ell=0$ and $i_\ell\ge1$, show that
$$P(z;k,\ell)=z^\ell P(z;k-1,\ell)+P(z;k,\ell-1).\tag{$\star$}$$
Then proceed by induction on $k+\ell\ge0$.
Solution. Such a partition $\mathcal P$ is uniquely determined by a $\ell$-tuple $(i_1,\ldots,i_\ell)$ where:
- for each $j\in\{1,\ldots,\ell\}$, $i_j\ge0$ indicates how many times part $j$ is repeated;
- $1i_1+2i_2+\cdots+\ell i_\ell=n$;
- $i_1+\cdots+i_\ell\le k$.
Therefore
\begin{align*}
P(z;k,\ell)\,
\quad
&=\sum_{i_1+\cdots+i_\ell\le k}z^{1i_1+2i_2+\cdots+\ell i_\ell}\\[.4em]
&=\quad z^\ell\sum_{\substack{i_1+\cdots+i_\ell\le k\\[.2em]i_\ell\ge1}}z^{1i_1+2i_2+\cdots +\ell(i_\ell-1)}
\quad+\sum_{\substack{i_1+\cdots+i_{\ell-1}\le k\\[.2em](i_\ell=0)}}z^{1i_1+2i_2+\cdots+(\ell-1)i_{\ell-1}}\\[.4em]
&=\quad z^\ell\sum_{i_1+\cdots+i_\ell\le k-1}z^{1i_1+2i_2+\cdots+\ell i_\ell}
\quad+\sum_{i_1+\cdots+i_{\ell-1}\le k}z^{1i_1+2i_2+\cdots+(\ell-1)i_{\ell-1}}\\[.4em]
&=z^\ell\,P(z;k-1,\ell)+P(z;k,\ell-1).\tag{$\star$}
\end{align*}
We now prove
$$k+\ell=s\implies
P(z;k,\ell)=\binom s{k,\ell}_z:=\dfrac{\prod\limits_{m=1}^s(1-z^m)}{\prod\limits_{i=1}^k(1-z^i)\cdot\prod\limits_{j=1}^\ell(1-z^j)}\tag{$\mathcal H_s$}$$
by induction on $s\ge0$.
We have $(\mathcal H_0)$ since $P(z;0,0)=1$ counts the empty partition. Let $s\ge1$ and suppose $(\mathcal H_{s-1})$.
Let $k,\ell\ge0$ with $k+\ell=s$.
Then
\begin{align*}
P(z;k,\ell)&\stackrel{(\star)}=z^\ell\,P(z;k-1,\ell)+P(z;k,\ell-1)\\[.4em]
&\!{\smash{\stackrel{(\mathcal H_{s-1})}=}}z^\ell\binom{s-1}{k-1,\ell}_z+\binom{s-1}{k,\ell-1}_z\\[.4em]
&\,=\left(\frac{1-z^s}{1-z^{s-\ell}}-\frac{1-z^\ell}{1-z^{s-\ell}}\right)\binom{s-1}{k-1,\ell}_z+\binom{s-1}{k,\ell-1}_z\\[.4em]
&\,=\frac{1-z^s}{1-z^k}\binom{s-1}{k-1,\ell}_z+\underbrace{\binom{s-1}{k,\ell-1}_z-\frac{1-z^\ell}{1-z^k}\binom{s-1}{k-1,\ell}_z}_{=\,0}\\[.4em]
&\,=\binom s{k,\ell}_z.
\end{align*}
Hence $(\mathcal H_s)$ is proved for all $s\ge0$.
Note. $(\star)$ is a $q$-analogue of Pascal's triangle formula. See Gaussian binomial coefficient.
