Poset probability in two-row partition posets

Mathematics > Combinatorics [Submitted on 19 Jan 2025 (v1), last revised 16 Jun 2026 (this version, v5)] Title:Poset probability in two-row partition posets View PDF HTML (experimental)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 \(\lambda_{1} - \lambda_{2}\). Submission history 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) Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.