Extending the science fiction and the Loehr--Warrington formula

We introduce the Macdonald piece polynomial ${\mathrm{I}}_{\mu ,\lambda ,k}[X;q,t]$, 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 ${\mathrm{I}}_{\mu ,\lambda ,k}$, $\mathrm{\nabla }{s}_{\lambda }$, and the Loehr–Warrington formula ${\mathrm{LW}}_{\lambda }$, thereby obtaining the Loehr–Warrington conjecture as a corollary. To connect ${\mathrm{I}}_{\mu ,\lambda ,k}$ and $\mathrm{\nabla }{s}_{\lambda }$, we employ the plethystic formula for the Macdonald polynomials of Garsia–Haiman–Tesler, and to connect ${\mathrm{I}}_{\mu ,\lambda ,k}$ and ${\mathrm{LW}}_{\lambda }$, 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 ${\mathrm{I}}_{\mu ,\lambda ,k}$.