Abstract
A famous problem in symmetric function theory is to find combinatorial formulas for Schur expansions of the Macdonald polynomials H ~ μ \tilde {H}_{\mu } . One such formula, valid for μ \mu satisfying μ 1 ≤ 3 \mu _1\leq 3 and μ 2 ≤ 2 \mu _2\leq 2 , involves Yamanouchi words weighted by Haglund’s statistics inv μ \operatorname {inv}_{\mu } and maj μ \operatorname {maj}_{\mu } . Previous proofs of this formula use the technical machinery of crystals and dual equivalence graphs. We give a new, elementary, and fully bijective proof of this formula based on the abacus model for antisymmetrized Macdonald polynomials. An extension to the Schur expansion of s ν H ~ μ s_{\nu }\tilde {H}_{\mu } is also provided.
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