Combinatorics in the Rational Shuffle Theorem and the Delta Conjecture

The Classical Shuffle Conjecture proposed by Haglund, Haiman, Loehr, Remmel and Ulyanov gives a well-studied combinatorial expression for the bigraded Frobenius characteristic of Sn-module of the ring of diagonal harmonics, which has been proved by Carlsson and Mellit as the Shuffle Theorem, stating that a symmetric function expression ∇en equals a generating function of combinatorial objects called parking functions. The Rational Shuffle Theorem of the expression Q_m,n(−1)^n of Mellit and the Delta Conjecture of the expression D'_ek en proposed by Haglund, Remmel and Wilson are two natural generalizations of the Shuffle Theorem. The primary goal of this dissertation is to prove some special cases of the conjectures, and compute the Schur function expansions of the corresponding symmetric function expressions. We explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of Q_m,n(−1)^n in the Rational Shuffle Theorem. Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of Q_m,n(−1)^n in the case where m or n equals 3. We give a combinatorial proof that the coefficient of s_lambda in the Delta expression D_e2 en has a non-negative expansion in terms of q,t-analogues. We propose a new valley version conjecture of the expression D'_ek D_hr en, and we give a proof of the valley version conjecture of D'_ek D_hr en when t or q equals 0. Our work lead to many new results about the combinatorial objects in the conjectures, such as the Mahonian distribution in extended ordered multiset partitions and the straightening action in parking functions