Crossings and nestings in tangled diagrams

W.Y.C. Chen, J. Qin, C.M. Reidys

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A tangled diagram on [n]={1,…,n}[n]={1,…,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 kk rows and kk-noncrossing tangled diagrams. We show that the numbers of kk-noncrossing and kk-nonnesting tangled diagrams are equal and we enumerate kk-noncrossing tangled diagrams. Finally, we show that braids, a special class of tangled diagrams, facilitate a bijection between 22-regular kk-noncrossing partitions and kk-noncrossing enhanced partitions.
Original languageEnglish
JournalThe Electronic Journal of Combinatorics
Publication statusPublished - 2008
Externally publishedYes


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