Svante Linusson: n! matchings, n! posets

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 uppertriangular matrices.

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

Välkomna!
A. Björner, A. Hultman, S. Linusson