Poset probability in two-row partition posets

Albin Jaldevik; Jan Snellman

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arXiv · 2501.11073

Poset probability in two-row partition posets

Albin Jaldevik·Jan Snellman

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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 $\lambda_{1} - \lambda_{2}$.

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Albin Jaldevik, Jan Snellman. 2025-01-19. Poset probability in two-row partition posets. https://arxiv.org/abs/2501.11073

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