Trivial Meet and Join within the Lattice of Monotone Triangles

Abstract

The lattice of monotone triangles (�n, ≼) 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 infimum τmin (maximul supremum τmax, resp.) is shown to asymptotically approach r|�n|r-1 as n→ ∞. Thus, with high probability this even implies that one of the τi is τmin (τmax, resp.). Higher-order error terms are also discussed