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 minimal infimum
τmin (maximal supremum τmax, resp.) is shown to
asymptotically approach r∣Mn∣r−1 as n→∞. Thus, with
high probability this event implies that one of the τi is τmin
(τmax, resp.). Higher-order error terms are also discussed.Comment: 15 page