Combinatorial Formulas for Macdonald and

This paper addresses a significant advancement in the theory of Macdonald polynomials, specifically focusing on the type A variation, where previous work by Haglund, Haiman, and Loehr introduced a combinatorial formula based on fillings of Young diagrams. The objective of this study is to demonstrate that this combinatorial formula, originally applicable to type A, can be derived more generally from the recent formula provided by Ram and Yip for Macdonald polynomials of arbitrary type, utilizing a technique known as compression. Additionally, this research extends the application of this framework to the Hall-Littlewood polynomials of type C, which emerge as specializations of their Macdonald counterparts when q is set to zero. This work also highlights the absence of a direct analogue to the Haglund-Haiman-Loehr formula for types beyond A, establishing the significance of our findings as a foundational step in exploring the existence of such formulas in broader contexts. The results emphasize the connection between different polynomial families and offer pathways for future research in algebraic combinatorics.