Generalization of the listed modulo-5, modulo-7, and modulo-11 congruences

Generalize the explicitly conjectured modulo-5, modulo-7, and modulo-11 congruences for \(\bar{b}^k_{r,s}(n)\) to other values of \(r\), \(s\), and \(k\).

Background

After listing computationally supported congruences for selected parameter triples and moduli 5, 7, and 11, the authors identify extending these congruences to additional values of the coloring parameters r,sr,s and tuple parameter kk as a problem for future work.

References

Lastly, we leave the problem of generalizing the above congruences for other values of $r,s$ and $k$ for future work.

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