The Electronic Journal of Combinatorics
The lattice of monotone triangles (𝕸n, ≼) ordered by entry-wise comparisons is studied. Let τmin denote the unique minimal element in this lattice, and τmax the unique maximum. The number of r-tuples of monotone triangles (τ1...,τr) with minimul infimumτmin (maximul supremum τmax, resp.) is shown to asymptotically approach r|𝕸n|r-1 asn→ ∞. Thus, with high probability this even implies that one of the τi is τmin (τmax, resp.). Higher-order error terms are also discussed.
Monotone triangle, alternating sign matrix, meet, join
Hammett, Adam J. and Engbers, John, "Trivial Meet and Join Within the Lattice of Monotone Triangles" (2014). Science and Mathematics Faculty Publications. 318.