Abstract:
I will present recent work joint with Anders Claesson. We study the class of matchings on the set $[2n]$ that contain no left-neighbor nesting. That is, matchings such that if $i$ is matched to $j$, $j>i$ and $i+1$ is matched to $k$, $k>i+1$ then $j<k$. We also define a class of naturally labeled $2+2$-free posets, called factorial posets. Bijections are given between both these sets of objects and permutations and hence they are both enumerated by $n!$. Our inspiration has come from the work of Bousquet-Mélou, Claesson, Dukes and Kitaev [arXiv:0806.0666] presented in our seminar in April by Claesson. It follows from our work that in their work one could replace "nesting" with "crossing". I will also show nice bijections between matchings with neighbor restrictions and certain upper triangular matrices.

I will state several conjectures concerning the distribution of patterns and enumeration of certain matchings.