Extending the science fiction and the Loehr--Warrington formula

We introduce the Macdonald piece polynomial Iμ,λ,k[X;q,t], which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between Iμ,λ,k, ∇ sλ, and the Loehr--Warrington formula LWλ, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect Iμ,λ,k and ∇ sλ, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect Iμ,λ,k and LWλ, we use our new findings on the combinatorics of P-tableaux together with the column exchange rule. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting Iμ,λ,k.