de Finetti Lattices and Magog Triangles

Abstract

Let B n,2 denote the order ideal of the boolean lattice B n consisting of all subsets of size at most 2. Let F n,2 denote the poset extension of B n,2 induced by the rule: i < j implies {i} ≺ {j} and {i, k} ≺ {j, k}. We give an elementary bijection from the set F n,2 of linear extensions of F n,2 to the set of shifted standard Young tableau of shape (n, n − 1, . . . , 1), which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set F(1) n,2 of poset extensions so that each singleton is comparable with all of the doubletons. We show that F(1) n,2 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.