Internal Congruences Modulo Powers of 5 for Partition k-Tuples with Odd Parts Distinct

Abstract

Let ${\text{pod}}_{-k}(n)$ denote the number of partition k-tuples of n where the odd parts in each partition are distinct. By utilizing some q-series manipulations and iterative computations, we derive dozens of internal congruences modulo powers of 5 for ${\text{pod}}_{-k}(n)$ with $1\le k\le 4$. For example, one result proved in the present paper is that for any $n\ge 0$ and $2\le m\le 12$, ${\text{pod}}_{-4}\left(5^{N_m}n+\frac{5^{N_m}+1}{2}\right)\equiv {\text{pod}}_{-4}\left(5^{m}n+\frac{5^m+1}{2}\right)\hspace{1em}(\mathrm{\text{mod}}{5}^m),$ where $N_m=2\times 5^{m-2}+m$. Further, we conjecture that these internal congruences exist in the corresponding internal congruence families modulo any powers of 5.

Mathematics Subject Classification:

• 11P83
• 05A17
• 05A15

Keywords:

• partitions
• k-tuples
• odd parts distinct
• internal congruences
• iterative computations
• generating functions

Acknowledgments

The author would like to acknowledge two anonymous referees for their careful reading and many helpful suggestions, which improved the quality of this paper to a great extent.

Declaration of Interest

The author reports there are no competing interests to declare.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (No. 12201093), the Natural Science Foundation Project of Chongqing CSTB (No. CSTB2022NSCQ–MSX0387), and the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202200509).