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-Melou and Viennot used for classical parallelogram polyominoes. We finally establish a connection between these new objects and 321-avoiding affine permutations. (C) 2018 Elsevier Inc. All rights reserved.
Biagioli R, Jouhet F, Nadeau P (2019). 321-avoiding affine permutations and their many heaps. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 162, 271-305 [10.1016/j.jcta.2018.11.002].
321-avoiding affine permutations and their many heaps
Biagioli R;
2019
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-Melou and Viennot used for classical parallelogram polyominoes. We finally establish a connection between these new objects and 321-avoiding affine permutations. (C) 2018 Elsevier Inc. All rights reserved.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
