Poset probability in two-row partition posets

We find explicit formulae for poset probabilities \(\mathbf{Prob}(P_\lambda; \alpha < \beta)\) in partition posets (cell posets) \(P_{\lambda}\) when \(\lambda=(\lambda_{1},\lambda_{2})\) is a two-row partition. These probabilities are given as rational expressions in \(f^{\sigma / \tau}\), where \(\tau \subseteq \sigma \subseteq \lambda\). We then use well-known formulae, such as the hook-length formula for \(f^\lambda\), the number of standard Young tableaux on a partition \(\lambda\), and the corresponding determinantal formula by Jacobi-Trudi-Aitken for \(f^{\lambda / \mu}\), the number of standard Young tableaux on a skew partition \(\lambda / \mu\), to make the aforementioned expressions explicit. We also calculate the limit probabilities of \(\mathbf{Prob}(P_\lambda; \alpha < \beta)\) when the elements \(\alpha,\beta\) are fixed cells, but the arm-lengths of \(\lambda=(\lambda_{1},\lambda_{2})\) tend to infinity with bounded difference \(\lambda_{1} - \lambda_{2}\).