Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations

TL;DR

This paper explores refined restrictions on ascent sequences and their effects on bijections with (2+2)-free posets, matrices, and permutations, revealing new combinatorial insights and duality effects.

Contribution

It introduces new restrictions on ascent sequences and analyzes their impact on associated combinatorial structures, including poset duality effects.

Findings

Identified natural and significant restrictions on ascent sequences.

Determined the impact of poset duality on related structures.

Provided deeper understanding of bijections among combinatorial objects.

Abstract

The combined work of Bousquet-M\'elou, Claesson, Dukes, Jel\'inek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent sequences, matrices and permutations, thereby answering a question of the first author and Parviainen in this case. The second restriction should appeal to Catalaniacs.