Trivial Meet and Join within the Lattice of Monotone Triangles

The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by entry-wise comparisons is studied. Let $τ_{\min}$ denote the unique minimal element in this lattice, and $τ_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(τ_1,\ldots,τ_r)$ with minimal infimum $τ_{\min}$ (maximal supremum $τ_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $τ_i$ is $τ_{\min}$ ($τ_{\max}$, resp.). Higher-order error terms are also discussed.

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