On the Subpartitions of the Ordinary Partitions, II

  • Byungchan Kim
  • Eunmi Kim
Keywords: partition, subpartition, partial theta function


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

\[\prod_{n=1}^{\infty} \frac{1}{1-q^n} \left( 1 - 2 \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \right).\]

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