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.
@article{10_37236_4049,
author = {John Engbers and Adam Hammett},
title = {Trivial meet and join within the lattice of monotone triangles.},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {3},
doi = {10.37236/4049},
zbl = {1325.06005},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4049/}
}
TY - JOUR
AU - John Engbers
AU - Adam Hammett
TI - Trivial meet and join within the lattice of monotone triangles.
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/4049/
DO - 10.37236/4049
ID - 10_37236_4049
ER -
%0 Journal Article
%A John Engbers
%A Adam Hammett
%T Trivial meet and join within the lattice of monotone triangles.
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/4049/
%R 10.37236/4049
%F 10_37236_4049
John Engbers; Adam Hammett. Trivial meet and join within the lattice of monotone triangles.. The electronic journal of combinatorics, Tome 21 (2014) no. 3. doi: 10.37236/4049
