N! matchings, n! posets (extended abstract)
2010 (English)In: FPSAC'10 - 22nd International Conference on Formal Power Series and Algebraic Combinatorics, 2010, 637-648 p.Conference paper (Refereed)
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-Mélou, 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-Mélou 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].
Place, publisher, year, edition, pages
2010. 637-648 p.
Ascent sequence, Crossing, Inversion table, Matching, Nesting, Permutation, Poset
IdentifiersURN: urn:nbn:se:kth:diva-150237ScopusID: 2-s2.0-84860505368OAI: oai:DiVA.org:kth-150237DiVA: diva2:745249
22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10, 2 August 2010 through 6 August 2010, San Francisco, CA, United States
QC 201409102014-09-102014-09-012014-09-10Bibliographically approved