Maximal supports and Schur-positivity among connected skew shapes

The Schur-positivity order on skew shapes is defined by B ≤ A if the difference s A − s B is Schur-positive. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. A strong necessary condition for the Schur-positivity of s A − s B is that the support of B is contained in that of A , where the support of B is defined to be the set of partitions λ for which s λ appears in the Schur expansion of s B . We show that to determine the maximal connected skew shapes in the Schur-positivity order and this support containment order, it suffices to consider a special class of ribbon shapes. We explicitly determine the support for these ribbon shapes, thereby determining the maximal connected skew shapes in the support containment order.