Date of Thesis

Spring 2019

Thesis Type

Honors Thesis

Degree Type

Bachelor of Science

Major

Mathematics

Second Major

Computer Science

Minor, Emphasis, or Concentration

Mathematics

First Advisor

Peter R. W. McNamara

Keywords

Generating function, partially ordered set, poset, fundamental basis, P-partition

Abstract

We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set's (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's 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.

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