Let us say a partition $\lambda$ is $Sylvestered$ if the smallest part to appear an even number of times ($0$ is an even number) is even. For each set of positive integers $T$, define $S(T\,;m,\,n)$ as the number of Sylvestered partitions of n into m parts such that no elements of $T$ are repeated. Then
$$\sum_{m;n \geq 0}S(T\, ; m,n)x^m q^n=\frac{\prod_{k\in T}(1-x^2 q^{2k})}{\prod_{j=1}^\infty (1-x q^j)}\sum_{n=1}^\infty \frac{x^{2n-1}q^{n(2n-1)}}{(-xq)_{2n}}$$,
and $(-xq)_{2n}=(1+xq)(1+xq^2) \cdots (1+xq^{2n})$.