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We study $321$-avoiding affine permutations, and prove a formula for theirenumeration with respect to the inversion number by using a combinatorialapproach. This is done in two different ways, both related to Viennot's theoryof heaps. First, we encode these permutations using certain heaps of monomersand dimers. This method specializes to the case of affine involutions. For thesecond proof, we introduce periodic parallelogram polyominoes, which are newcombinatorial objects of independent interest. We enumerate them by extendingthe approach of Bousquet-M\'elou and Viennot used for classical parallelogrampolyominoes. We finally establish a connection between these new objects and$321$-avoiding affine permutations.

321-avoiding affine permutations and their many heaps