Geodesic
Crossings and nestings in tangled diagrams
The electronic journal of combinatorics, Tome 15 (2008)

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A tangled diagram on $[n]=\{1,\dots,n\}$ is a labeled graph for which each vertex has degree at most two. The vertices are arranged in increasing order on a horizontal line and the arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Generalizing the construction of Chen et al., we give a bijection between generalized vacillating tableaux with less than $k$ rows and $k$-noncrossing tangled diagrams. We show that the numbers of $k$-noncrossing and $k$-nonnesting tangled diagrams are equal and we enumerate $k$-noncrossing tangled diagrams. Finally, we show that braids, a special class of tangled diagrams, facilitate a bijection between $2$-regular $k$-noncrossing partitions and $k$-noncrossing enhanced partitions.
DOI : 10.37236/810
Classification : 05A18
Mots-clés : tangled diagram, labeled graph, vacillating tableaux 
William Y. C. Chen; Jing Qin; Christian M. Reidys. Crossings and nestings in tangled diagrams. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/810
@article{10_37236_810,
     author = {William Y. C. Chen and Jing Qin and Christian M. Reidys},
     title = {Crossings and nestings in tangled diagrams},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/810},
     zbl = {1163.05309},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/810/}
}
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