We revisit the Rédei-Berge symmetric function $\mathcal{U}_D$ for digraphs $D$, a specialization of Chow's path-cycle symmetric function. Through the lens of matrix algebra, we consolidate and expand on the work of Chow, Grinberg and Stanley, and Lass concerning the resolution of $\mathcal{U}_D$ in the power sum and Schur bases. Along the way we also revisit various results on Hamiltonian paths in digraphs.