(2+2)-free posets, ascent sequences and pattern avoiding permutations

TL;DR

This paper establishes bijections between four combinatorial classes, including (2+2)-free posets, involutions, pattern-avoiding permutations, and ascent sequences, revealing their enumerative properties and solving a conjecture.

Contribution

It introduces new bijections among four classes of combinatorial objects, including a direct bijection between (2+2)-free posets and involutions, and characterizes ascent sequences related to pattern-avoiding permutations.

Findings

Derived the generating function for the classes, recovering Zagier's non-D-finite series.

Established a bijection preserving statistics between posets and involutions.

Solved Pudwell's conjecture by enumerating permutations avoiding a specific barred pattern.

Abstract

We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2+2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of $D_8$, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2+2)-free posets and permutations. Our bijections preserve numerous statistics. We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3{\bar 1}52{\bar 4}$ and use this to enumerate those permutations, thereby settling a conjecture of Pudwell.