Abstract: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 \(\lambda_{1} - \lambda_{2}\).
From: Jan Snellman [view email]
[v1]
Sun, 19 Jan 2025 15:21:09 UTC (300 KB)
[v2]
Thu, 23 Jan 2025 06:50:10 UTC (351 KB)
[v3]
Mon, 23 Jun 2025 11:53:46 UTC (3,561 KB)
[v4]
Fri, 4 Jul 2025 13:10:55 UTC (3,562 KB)
[v5]
Tue, 16 Jun 2026 14:21:45 UTC (3,368 KB)
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