Haglund's conjecture on 3-column Macdonald polynomials

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund (Proc Natl Acad Sci USA 101(46):16127-16131, 2004). The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's (Eur J Combin 29(1):343-359, 2008) algebra of ribbon Schur operators. Combining this result with the expression of Haglund et al. (J Am Math Soc 18(3):735-761, 2005) for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.