In this note, we provide a new proof for the number of partitions of $n$ having subpartitions of length $\ell$ with gap $d$. Moreover, by generalizing the definition of a subpartition, we show what is counted by $q$-expansion \[ \prod_{n=1}^{\infty} \frac{1}{1-q^n} \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \] and how fast it grows. Moreover, we prove there is a special sign pattern for the coefficients of $q$-expansion \[ \prod_{n=1}^{\infty} \frac{1}{1-q^n} \left( 1 - 2 \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \right). \]