  We develop combinatorial tools to study the relationship between the Stanley
depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$.
Using these results we are able to prove that if $S$ is a polynomial ring with
at most 5 indeterminates and $I$ is a square-free monomial ideal, then the
Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using
a computer search, we are able to extend this strict inequality up to
polynomial rings with at most 7 indeterminates. This partially answers
questions asked by Propescu and Qureshi as well as Herzog.
