Geodesic
On the symmetry of the distribution of \(k\)-crossings and \(k\)-nestings in graphs
The electronic journal of combinatorics, Tome 13 (2006)
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This note contains two results on the distribution of $k$-crossings and $k$-nestings in graphs. On the positive side, we exhibit a class of graphs for which there are as many $k$-noncrossing $2$-nonnesting graphs as $k$-nonnesting $2$-noncrossing graphs. This class consists of the graphs on $[n]$ where each vertex $x$ is joined to at most one vertex $y$ with $y < x$. On the negative side, we show that this is not the case if we consider arbitrary graphs. The counterexample is given in terms of fillings of Ferrers diagrams and solves a problem of Krattenthaler.
DOI : 10.37236/1159
Classification : 05C10 
@article{10_37236_1159,
     author = {Anna de Mier},
     title = {On the symmetry of the distribution of \(k\)-crossings and \(k\)-nestings in graphs},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1159},
     zbl = {1114.05027},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1159/}
}
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Anna de Mier. On the symmetry of the distribution of \(k\)-crossings and \(k\)-nestings in graphs. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1159

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