Combinatorial formulas for symmetric Macdonald polynomials by superizations

In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\lambda }(X;q,t)$ and integral Macdonald polynomials $J_{\lambda }(X;q,t)$, in terms of several new statistics and the major index for a partition $\lambda$. Compared to previous formulas, these new ones contain the fewest terms. Moreover, three existing formulas for symmetric Macdonald polynomials established by Corteel–Mandelshtam–Williams (2022), Corteel–Haglund–Mandelshtam–Mason–Williams (2022) and Mandelshtam (2025) are recovered. Our proof relies on a new statistic on super fillings, employing the superization formulas of Haglund–Haimain–Loehr (2005) and Ayyer–Mandelshtam–Martin (2023), together with our recent approach to modified Macdonald polynomials.