A Combinatorial Formula for the Schur Coefficients of the Integral Form of the Macdonald Polynomials in the Two Column and Certain Hook Cases

In this paper, we prove that the integral form of Macdonald polynomials J _?? [X; q, t] has the property that J _?? [X; q, t]/(1?q) ⁿ has Schur expansion with positive polynomial coefficient. Our proof proceeds by constructing constructing combinatorial formula for the Schur coefficients when??? is either a two column shape or a certain type of hook shape.