Explicit Plethystic Formulas for Macdonald q,t-Kostka Coefficients

i=1 ti−1 (1+· · ·+qμi−1). In [8] Garsia-Tesler proved that if γ is a partition of k and λ = (n−k, γ) is a partition of n, then there is a unique symmetric polynomial kγ(x; q, t) of degree ≤ k with the property that K̃λμ(q, t) = kγ [Bμ(q, t); q, t] holds true for all partitions μ. It was shown there that these polynomials have Schur function expansions of the form kγ(x; q, t) = ∑ |ρ|≤|γ| Sλ(x) kρ,γ(q, t) where the kρ,γ(q, t) are polynomials in q, t, 1/q, 1/t with integer coefficients. This result yielded the first proof of the Macdonald polynomiality conjecture. It also was used in a proof [7] of the positivity conjecture for the K̃λμ(q, t) for any λ of the form λ = (r, 2, 1) and arbitrary μ. In this paper we show that the polynomials kγ(x; q, t) may be given a very simple explicit expression in terms of the operator ∇ studied in [2]. In particular we also obtain a new proof of the polynomiality of the coefficients K̃λμ(q, t). Further byproducts of these developments are a new explicit formula for the polynomial H̃μ[X; q, t] = ∑ λ Sλ[X]K̃λμ(q, t) and a new derivation of the symmetric function results of Sahi [16] and Knop [11], [12].