The strong Hall property and symmetric chain orders Original Research Article

Let G=(X,Y;E) be a bipartite graph with |X|⩾|Y|. For A⊆X, write φ(A)=|A|−|N(A)| and for a⩽|X|, define φ(a)=max{φ(A) | A⊆X, |A|=a}. The graph G is said to have the strong Hall property if φ(a)+φ(b)⩽|X|−|Y| for all nonnegative integers a and b with a+b⩽|X|. We shall prove that any unimodal and self-dual poset with the strong Hall property is a symmetric chain order. This result will also be used to show that the inversion poset S5 is a symmetric chain order.