The $k$-elongated plane partition function modulo small powers of $5$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909579858673664 |
|---|---|
| author | Guadalupe, Russelle |
| author_facet | Guadalupe, Russelle |
| contents | Andrews and Paule revisited combinatorial structures known as the $k$-elongated partition diamonds, which were introduced in connection with the study of the broken $k$-diamond partitions. They found the generating function for the number $d_k(n)$ of partitions obtained by summing the links of such partition diamonds of length $n$ and discovered congruences for $d_k(n)$ using modular forms. Since then, congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary means and modular forms by many authors, most recently Banerjee and Smoot who established an infinite family of congruences for $d_5(n)$ modulo powers of $5$. We extend in this paper the list of known results for $d_k(n)$ by proving infinite families of congruences for $d_k(n)$ modulo $5,25$, and $125$ using classical $q$-series manipulations and $5$-dissections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08627 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $k$-elongated plane partition function modulo small powers of $5$ Guadalupe, Russelle Number Theory Combinatorics 11P83, 05A17 Andrews and Paule revisited combinatorial structures known as the $k$-elongated partition diamonds, which were introduced in connection with the study of the broken $k$-diamond partitions. They found the generating function for the number $d_k(n)$ of partitions obtained by summing the links of such partition diamonds of length $n$ and discovered congruences for $d_k(n)$ using modular forms. Since then, congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary means and modular forms by many authors, most recently Banerjee and Smoot who established an infinite family of congruences for $d_5(n)$ modulo powers of $5$. We extend in this paper the list of known results for $d_k(n)$ by proving infinite families of congruences for $d_k(n)$ modulo $5,25$, and $125$ using classical $q$-series manipulations and $5$-dissections. |
| title | The $k$-elongated plane partition function modulo small powers of $5$ |
| topic | Number Theory Combinatorics 11P83, 05A17 |
| url | https://arxiv.org/abs/2504.08627 |