  We consider the skew diagram $\Delta_n$, which is the $180^\circ$ rotation of
the staircase diagram $\delta_n = (n,n-1,n-2,...,2,1)$. We create a staircase
with bad foundation by augmenting $\Delta_n$ with another skew diagram, which
we call the \textit{foundation}. We consider pairs of staircases with bad
foundation whose foundations are transposes of one another. Among these pairs,
we show that the difference of the corresponding skew Schur functions is
Schur-positive in the case when one of the foundations consists of either a one
or two row diagram, or a hook diagram.

Combinatorics