  The paper compares (and reproves) the alcove walk and the nonattacking
fillings formulas for type $GL_n$ Macdonald polynomials which were given in
Haglund-Haiman-Loehr arXiv:math.CO/0601693, Alexandersson arXiv:1602.05153 and
Ram-Yip arXiv:0803.1146. The "compression" relating the two formulas in this
paper is the same as that of Lenart arXiv:0804.4716. We have reformulated it so
that it holds without conditions and so that the proofs of the alcove walk
formula and the nonattacking fillings formula are parallel. This reformulation
highlights the role of the double affine Hecke algebra and Cherednik's
intertwiners. An exposition of the type $GL_n$ double affine braid group,
double affine Hecke algebra, and all definitions and proofs regarding Macdonald
polynomials are provided to make this paper self contained.
