Séminaire Lotharingien de Combinatoire, 93B.2 (2025), 12 pp.
Donghyun Kim and Jaeseong Oh
Extending the Science Fiction and the Loehr-Warrington Formula
Abstract.
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. We also present an extension of the science
fiction conjecture and the Macdonald positivity by exploiting
Iμ,λ,k.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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