On $k$-noncrossing partitions

In this paper we prove a duality between $k$-noncrossing partitions over $[n]=\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ and $k$-noncrossing braids over $[n-1]$. This duality is derived directly via (generalized) vacillating tableaux which are in correspondence to tangled-diagrams \cite{Reidys:07vac}. We give a combinatorial interpretation of the bijection in terms of the contraction of arcs of tangled-diagrams. Furthermore it induces by restriction a bijection between $k$-noncrossing, 2-regular partitions over $[n]$ and $k$-noncrossing braids without isolated points over $[n-1]$. Since braids without isolated points correspond to enhanced partitions this allows, using the results of \cite{MIRXIN}, to enumerate 2-regular, 3-noncrossing partitions.