  We summarize the known useful and interesting results and formulas we have
discovered so far in this collaborative article summarizing results from two
related articles by Merca and Schmidt arriving at related so-termed Lambert
series factorization theorems. We unify the matrix representations that
underlie two of our separate papers, and which commonly arise in identities
involving partition functions and other functions generated by Lambert series.
We provide a number of properties and conjectures related to the inverse matrix
entries defined in Schmidt's article and the Euler partition function $p(n)$
which we prove through our new results unifying the expansions of the Lambert
series factorization theorems within this article.
