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\[\prod_{n=1}^{\infty} \frac{1}{1-q^n} \left( 1 - 2 \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \right).\]
@article{10_37236_3526,
author = {Byungchan Kim and Eunmi Kim},
title = {On the subpartitions of the ordinary partitions. {II}},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/3526},
zbl = {1302.05009},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3526/}
}
TY - JOUR
AU - Byungchan Kim
AU - Eunmi Kim
TI - On the subpartitions of the ordinary partitions. II
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/3526/
DO - 10.37236/3526
ID - 10_37236_3526
ER -
%0 Journal Article
%A Byungchan Kim
%A Eunmi Kim
%T On the subpartitions of the ordinary partitions. II
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/3526/
%R 10.37236/3526
%F 10_37236_3526
Byungchan Kim; Eunmi Kim. On the subpartitions of the ordinary partitions. II. The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/3526
