A character approach to directed genus distribution of graphs: The bipartite single-black-vertex case

Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such is log-concave for two families digraphs, thus giving a positive answer these to question asked in Bonnington, Conder, Morton and McKenna (2002). Our proof uses real-rooted polynomials representation theory symmetric group $\mathbb{S}_n$. The result also extended some factorizations identity $\mathbb{S}_n$ are rotation systems one-face constellations.