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Abstract

This work studies combinatorics related to expansions of a quasisymmetric refinement of Schur functions into Gessel’s fundamental basis, with almost all results concerning whether an expansion is multiplicity-free. The combinatorial side of this problem concerns a certain composition poset, and whether there are two standard fillings of the same composition diagram with a given descent set. This thesis uses entirely combinatorial arguments, extending results by Bessenrodt and vanWilligenburg to work towards a classification of such descent multiplicity-free compositions. The main tools used regard the situation of appending or prepending parts to compositions. Compositions with multiplicity retain multiplicity when parts are appended or prepended, while multiplicity-free compositions stay multiplicity-free when a class of shapes called staircase-like are appended. A classification of compositions which are partitions or reverse partitions is achieved, leading up to a classification of compositions not containing a part of length one. This is used as the basis for a conjectured classification of multiplicity-free compositions without a trailing staircase. The conjecture would in turn imply a complete classification of multiplicity-free compositions.