Let \(\overline{NT}(r, m, n)\) count the total number of parts in overpartitions of n with rank congruent to \(r \pmod {m}\) . Recently, Dan, Liu, and Yao proved the following Andrews–Beck type congruence for overpartitions using theta functions: for all \(n \ge 0\) \(\begin{aligned} \sum _{r = 1}^{3} r \overline{NT}(r, 4, n) = {\left\{ \begin{array}{ll} 2 \pmod 4, & \text {if }\,\, n = 2k^{2} \,\, \text { or }\,\, n = 4k^{2} \,\,\text { for some}\,\, k \in \mathbb {Z^{+}}, \\ 0 \pmod 4, & \text { otherwise.} \end{array}\right. } \end{aligned}\) They posed the problem of finding a combinatorial proof of this congruence. In this paper, we provide such a combinatorial proof.
