Weingarten calculus provides a representation-theoretic framework for evaluating Haar integrals of products of matrix entries over compact groups. In this talk, I will present a polynomial method for $U_N$ and $SU_N$ Weingarten calculus, focusing on contingency tables, Kostka operators, and polarization. Generating polynomials encode matrix-entry integrals, while contingency tables organize commutative monomials and lead to monomial integral formulas through Kostka-type operators. Polarization then restores the ordered information needed to recover link integrals from the same generating polynomial. I will also explain how the $SU_N$ theory differs from the $U_N$ theory: the determinant-one condition introduces shifted matching conditions and determinant powers in the character expansion. Finally, I will discuss an application to the Alon--Tarsi conjecture, where the special determinant type $SU_N$ integral recovers the signed difference between even and odd Latin squares and leads to new combinatorial interpretations through rectangular symmetric-group characters and permutation factorizations.