Congruence families for progressions 27n+9 and 27n+18

Investigate and derive congruence families for the overcolored partition k-tuple numbers \(\bar{b}^k_{r,s}(n)\) in the arithmetic progressions \(27n+9\) and \(27n+18\), extending the families established for progressions \(3n+1\), \(3n+2\), \(9n+3\), and \(9n+6\).

Background

The paper proves several infinite families of congruences modulo 3 for bˉr,sk(n)\bar{b}^k_{r,s}(n), including congruences in progressions $3n+1$, $3n+2$, $9n+3$, and $9n+6$. It then states one further collection of congruences for progressions $27n+9$ and $27n+18$, but explains that obtaining broader families analogous to the preceding theorems would require substantially more elaborate computations.

References

It is natural to expect an extension of these results to other families. However, deriving families similar to Theorems \ref{6tmod3.2}-\ref{6tmod3.4} for progressions $27n+9$ and $27n+18$ requires considerably more elaborate computations. We therefore leave investigation of such congruence families as an open problem.

— Overcolored Partition $k$-tuples Restricted by Parity of the Parts  (2609.03926 - Thejitha et al., 3 Sep 2026) in Section 6s7, Concluding remarks, item 1