Positivity Among P-Partition Generating Functions of Partially Ordered Sets

We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set\u27s (poset) partition generating functions to be positive in the fundamental basis. We define the notion of a jump sequence for a poset and show how different conditions on the jump sequences of two posets are necessary for those posets to have an order relation in the fundamental basis. Our sufficient conditions are of two types. First, we show how manipulating a poset\u27s Hasse diagram produces a poset that is greater according to the fundamental basis. Secondly, we also provide tools to explain posets that are constructed by combining other posets in certain ways through the so-called Ur-operation. Finally, we are able to provide both necessary and sufficient conditions for positivity among posets of Greene shape (k,1) and among a subclass of caterpillar posets, and a complete (and graphically pleasing) representation of the order relations between posets of the former type