Dissecting the Stanley partition function

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:|p0(2n)-p2(2n)|>|p0(2n+1)-p2(2n+1)|,n>0.Two proofs of this surprising inequality are given. The first one uses the Göllnitz-Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.