Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$

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.