  We define a fat staircase to be a Ferrers diagram corresponding to a
partition of the form $(n^{\alpha_n}, {n-1}^{\alpha_{n-1}},..., 1^{\alpha_1})$,
where $\alpha = (\alpha_1,...,\alpha_n)$ is a composition, or the $180^\circ$
rotation of such a diagram. If a diagram's skew Schur function is a linear
combination of Schur functions of fat staircases, we call the diagram a sum of
fat staircases. We prove a Schur-positivity result that is obtained each time
we augment a sum of fat staircases with a skew diagram. We also determine
conditions on which diagrams can be sums of fat staircases, including necessary
and sufficient conditions in the special case when the diagram is a fat
staircase skew a single row or column.
