Conjectural modulo-5 congruences for overcolored partition k-tuples

Prove the conjectured modulo-5 congruences \(\bar{b}^4_{4,2}(5n)\), \(\bar{b}^4_{5,3}(5n)\), \(\bar{b}^2_{2,4}(5n)\), \(\bar{b}^2_{3,4}(5n)\), \(\bar{b}^4_{4,4}(5n)\), \(\bar{b}^3_{5,4}(5n)\), \(\bar{b}^4_{1,2}(5n)\), and \(\bar{b}^8_{1,3}(5n)\) for all \(n\ge1\).

Background

The authors report computational evidence for several congruences modulo 5 involving the coefficients bˉr,sk(n)\bar{b}^k_{r,s}(n) associated with overcolored partition k-tuples. These relations are explicitly presented as conjectures rather than proved results.

References

Based on computational evidences, we list few congruences modulo $5,7$, and $11$ analogues to those in Theorem 4.1-4.3 , in the following conjectures. \begin{conjecture} For all $n\ge1$, \bar{b}4_{4,2}(5n)\equiv 0\pmod{5},\quad \bar{b}4_{5,3}(5n)\equiv 0\pmod{5},\quad \bar{b}2_{2,4}(5n)\equiv 0\pmod{5},\quad \bar{b}2_{3,4}(5n)\equiv 0\pmod{5},\quad \bar{b}4_{4,4}(5n)\equiv 0\pmod{5},\quad \bar{b}3_{5,4}(5n)\equiv 0\pmod{5},\quad \bar{b}4_{1,2}(5n)\equiv 0\pmod{5},\quad \bar{b}8_{1,3}(5n)\equiv 0\pmod{5}. \end{conjecture}

— 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, first conjecture