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n! matchings, n! posets

Claesson, Anders and Linusson, Svante (2011) n! matchings, n! posets. Proceedings of the American Mathematical Society, 139 (2). 435–449. ISSN 0002-9939

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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]

Item type: Article
ID code: 34549
Keywords: (2+2)-free poset, ascent sequences, bijections, Mathematics
Subjects: Science > Mathematics
Department: Faculty of Science > Computer and Information Sciences
Related URLs:
    Depositing user: Pure Administrator
    Date Deposited: 25 Oct 2011 09:57
    Last modified: 14 Dec 2012 12:56
    URI: http://strathprints.strath.ac.uk/id/eprint/34549

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