Trivial Meet and Join within the Lattice of Monotone Triangles

Abstract

The lattice of monotone triangles $({\mathfrak{M}}_n,\le )$ 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 $({\tau }_1,\dots ,{\tau }_r)$ 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.

15 pages