321-avoiding affine permutations and their many heaps

TL;DR

This paper provides a combinatorial enumeration of 321-avoiding affine permutations using heaps and introduces new periodic parallelogram polyominoes, establishing a connection between these objects and the permutations.

Contribution

It introduces two novel combinatorial methods for counting 321-avoiding affine permutations and defines new polyominoes linked to these permutations.

Findings

Derived a formula for enumeration by inversion number

Encoded permutations using heaps of monomers and dimers

Established a connection with new periodic parallelogram polyominoes

Abstract

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