Factorizations of Pieri rules for Macdonald polynomials Original Research Article

We introduce a heuristic embedding of the Macdonald polynomials Pμ(x; q, t) into a family of polynomials indexed by lattice square diagrams. This embedding leads to recursions which may be viewed as a factorization of the Stanley-Macdonald (1988, 1989) Pieri rules and shed some light into their intricate nature. In this manner we can prove some conjectures concerning the coefficients Kλμ(q, t). In particular, when μ is a 2-row shape or a hook we show that the expression ∑λƒλKλμ(q, t) is a polynomial with nonnegative integer coefficients. The recursions obtained in these cases lead to a combinatorial interpretation of this polynomial as a q, t-enumerator of permutations. A new proof is also obtained of a Jacobi-Trudi identity for 2-row shapes recently obtained by Jing and Jósefiak. Some examples involving more general shapes are also included.