De Finetti Lattices and Magog Triangles

Let $F$ be the collection of subsets of $[n]={1,2,…,n}$ of size at most 2 with the partial order rule $i<j$ implies ${i}≺{j}$ and ${i,k}≺{j,k}$. We show that the number of linear extensions of $F$ is counted by the strict sense ballot numbers. More surprisingly, we show that the number of partial orders so that all singletons are comparable with all doubletons is counted by the Robbins numbers. which is also the number of alternating sign matrices.