n! matchings, n! posets

TL;DR

This paper demonstrates that the number of matchings without neighbor nestings on 2n points and the number of naturally labeled (2+2)-free posets on n elements are both n!, establishing a combinatorial equivalence and exploring related bijections and statistics.

Contribution

It introduces a new set of labeled (2+2)-free posets, proves their enumeration matches that of certain matchings, and constructs bijections linking nesting and crossing restrictions.

Findings

Number of matchings without neighbor nestings is n!

Number of labeled (2+2)-free posets is n!

Bijections between nesting and crossing restrictions are established

Abstract

We show that there are $n!$ matchings on $2n$ points without, so called, left (neighbor) nestings. We also define a set of naturally labeled $(2+2)$-free posets, and show that there are $n!$ such posets on $n$ elements. Our work was inspired by Bousquet-M\'elou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884--909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabeled $(2+2)$-free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-M\'elou et al.\ and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled $(2+2)$-free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) \#R53]