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.