This talk presents a polynomial method for studying Haar integrals over $U_N$ and $SU_N$, motivated by matrix-model partition functions and link integrals in lattice gauge theory. Instead of evaluating ordered matrix-entry integrals through invariant tensor bases or entrywise differentiation, we package them into generating polynomials and organize their coefficients using polarization, commutative monomials, and contingency tables. This leads to the Kostka-operator formula for $SU_N$ monomial integrals. As an application, we relate a one-sided $SU_N$ integral to the coefficient of the full-support monomial in $(\det X)^N$, which equals the difference between the numbers of even and odd Latin squares, giving a new combinatorial perspective on the Alon--Tarsi conjecture.