On the Maximal Containments of Lower Central Series Ideals

We begin a discussion about the maximal containments of lower central series ideals: ideals generated by products of two-sided ideals of the lower central series of the free associative algebra on $n$ generators. We introduce two new ideas to the topic, the PBW grading and the pigeonhole principle, that help us give a complete classification of the containments for $n=2$ and obtain partial results in the general case.