The $(km,kn)$-square paths theorem

For positive integers $k,m,n$ with $\gcd (m,n)=1$, we prove a $(km,kn)$-square paths theorem: the symmetric function $(-1{)}^{k-1}\frac{[km{]}_q}{[k{]}_q}{p}_k[-M{X}^{m,n}]\cdot 1$ is a weighted sum over $(km,kn)$-preference functions. This proves a corrected form of the conjecture of Iraci, Pagaria, Paolini, and Vanden Wyngaerd and generalizes the square paths theorem of Sergel. We encode preference functions by row strict tableaux and closed walks on a set of states, compare them with parking functions for each value of the touch statistic, and apply Mellit’s compositional $(km,kn)$-shuffle theorem. Finally, we establish a fall Delta square theorem conjectured by Iraci, Pagaria, and Paolini.