UC San Diego

We conjecture two combinatorial interpretations for the symmetric function Delta/ek en, where [Delta]f is an eigenoperator for the modified Macdonald polynomials defined by Garsia and Haiman. The first interpretation is due to Haglund and the second is due to the author. Both interpretations can be seen as generalizations of the Shuffle Conjecture of Haglund, Haiman, Remmel, Loehr, and Ulyanov. The primary goal of this dissertation is to prove various special cases of these conjectures. We accomplish this goal by connecting the interpretations to objects such as ordered set partitions, rook placements, Tesler matrices, and LLT polynomials. These connections lead to many new results about these objects, such as an extension of MacMahon's classical equidistribution theorem from permutations to ordered set partitions