Trivial meet and join within the lattice of monotone triangles. (Q405297)

Summary: The lattice of monotone triangles \((\mathfrak M_n,\leq)\) ordered by entry-wise comparisons is studied. Let \(\tau_{\min}\) denote the unique minimal element in this lattice, and \(\tau_{\max}\) the unique maximum. The number of \(r\)-tuples of monotone triangles \((\tau_1,\ldots,\tau_r)\) with minimal infimum \(\tau_{\min}\) (maximal supremum \(\tau_{\max}\), 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 \(\tau_i\) is \(\tau_{\min}\) (\(\tau_{\max}\), resp.). Higher-order error terms are also discussed.