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Differences of Skew Schur Functions of Staircases with Transposed Foundations

We consider the skew diagram $Δ_n$, which is the $180^\circ$ rotation of the staircase diagram $δ_n = (n,n-1,n-2,...,2,1)$. We create a staircase with bad foundation by augmenting $Δ_n$ 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.