  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. The proof requires expressing a noncommutative Schur function as a
positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining
this result with the expression of Haglund, Haiman, and Loehr 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.
