Poset probability in two-row partition posets

Abstract

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