Two product formulas for counting successive vertex orderings

A vertex ordering of a graph $G$ is a bijection $\pi \text{\hspace{0.17em}⁣}:\{1,\dots ,\mathrm{\mid }V(G)\mathrm{\mid }\}\to V(G)$. It is successive if the induced subgraph $G[v_{\pi (1)},\dots ,v_{\pi (k)}]$ is connected for each $k$. Lixing Fang, Hao Huang, János Pach, Gábor Tardos, and Junchi Zuo [J. Comb. Theory A199 (2023), 105776] gave formulas for counting the number of successive vertex orderings for a class of graphs they called “fully regular,” and conjectured that these formulas could be written as certain products involving differences or ratios of binomial coefficients in two cases: When the graph is the line graph $L(K_n^{(3)})$ of the complete $3$-uniform hypergraph, or when it is the line graph $L(K_{m,n}^{(1,2)})$ of a complete “bipartite” $3$-uniform hypergraph. In this paper, we confirm both of these conjectures.