The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.
@article{10_37236_9246,
author = {Andrew Beveridge and Ian Calaway and Kristin Heysse},
title = {De {Finetti} lattices and magog triangles},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {1},
doi = {10.37236/9246},
zbl = {1458.05011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9246/}
}
TY - JOUR
AU - Andrew Beveridge
AU - Ian Calaway
AU - Kristin Heysse
TI - De Finetti lattices and magog triangles
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/9246/
DO - 10.37236/9246
ID - 10_37236_9246
ER -
%0 Journal Article
%A Andrew Beveridge
%A Ian Calaway
%A Kristin Heysse
%T De Finetti lattices and magog triangles
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/9246/
%R 10.37236/9246
%F 10_37236_9246
Andrew Beveridge; Ian Calaway; Kristin Heysse. De Finetti lattices and magog triangles. The electronic journal of combinatorics, Tome 28 (2021) no. 1. doi: 10.37236/9246
