논문
On the subpartitions of the ordinary partitions, II
- 저자Byungchan Kim, Eunmi Kim.
- 학술지Electronic Journal of Combinatorics 21(4)
- 등재유형
- 게재일자(2014)
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
\[
\prod_{n=1}^{\infty} \frac{1}{1-q^n} \left( 1 - 2 \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \right).
\]
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
\[
\prod_{n=1}^{\infty} \frac{1}{1-q^n} \left( 1 - 2 \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \right).
\]
