Let $b_\ell(n)$ denote the number of $\ell$-regular partitions of $n$. Congruences properties modulo powers of $2$ for $b_4(n)$ were considered subsequently by Andrews-Hirschhorn-Sellers, Chen, Cui-Gu, Xia, Dai, and Ballantine-Merca. In this paper, we present an approach which can be utilized to prove the ``self-similar'' congruence property satisfied by the generating function of $b_4(n)$. As an immediate consequence, one can obtain dozens of congruence families modulo powers of $2$ enjoyed by $b_4(n)$. These results not only generalize some previous results, but also can be viewed as a supplement to Keith and Zanello's comprehensive study of the congruence properties for $\ell$-regular partition functions. Finally, we also pose several conjectures on congruence families, internal congruence families and self-similar congruence properties for $4$-, $8$- and $16$-regular partition functions.
@article{10_37236_11919,
author = {Julia Q. D. Du and Dazhao Tang},
title = {Congruence properties modulo powers of \(2\) for \(4\)-regular partitions},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {3},
doi = {10.37236/11919},
zbl = {1559.11107},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11919/}
}
TY - JOUR
AU - Julia Q. D. Du
AU - Dazhao Tang
TI - Congruence properties modulo powers of \(2\) for \(4\)-regular partitions
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/11919/
DO - 10.37236/11919
ID - 10_37236_11919
ER -
%0 Journal Article
%A Julia Q. D. Du
%A Dazhao Tang
%T Congruence properties modulo powers of \(2\) for \(4\)-regular partitions
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/11919/
%R 10.37236/11919
%F 10_37236_11919
Julia Q. D. Du; Dazhao Tang. Congruence properties modulo powers of \(2\) for \(4\)-regular partitions. The electronic journal of combinatorics, Tome 31 (2024) no. 3. doi: 10.37236/11919
