De Finetti Lattices and Magog Triangles

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists all subsets size at most $2$. Let $F_{n,2}$ denote poset refinement induced by rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \{j,k\}$. We give an elementary bijection from set $\mathcal{F}_{n,2}$ linear extensions to shifted standard Young tableau shape $(n, n-1, \ldots, 1)$, which are counted strict-sense ballot numbers. find a more surprising result when considering $\mathcal{F}_{n,2}^{1}$ minimal refinements in each singleton is comparable with doubletons. show that magog triangles, therefore equinumerous alternating sign matrices. adopt our proof techniques row reversal matrix corresponds natural involution on gog triangles.