Variants of the RSK algorithm adapted to combinatorial Macdonald polynomials

Abstract

We introduce variations of the Robinson–Schensted correspondence parametrized by positive integers p. Each variation gives a bijection between permutations and pairs of standard tableaux of the same shape. In addition to sharing many of the properties of the classical Schensted algorithm, the new algorithms are designed to be compatible with certain permutation statistics introduced by Haglund in the study of Macdonald polynomials. In particular, these algorithms provide an elementary bijective proof converting Haglund's combinatorial formula for Macdonald polynomials to an explicit combinatorial Schur expansion of Macdonald polynomials indexed by partitions μ satisfying μ1≤3 and μ2≤2. We challenge the research community to extend this RSK-based approach to more general classes of partitions.