  We construct an injection from the set of $r$-fans of Dyck paths (resp.
vacillating tableaux) of length $n$ into the set of chord diagrams on $[n]$
that intertwines promotion and rotation. This is done in two different ways,
namely as fillings of promotion-evacuation diagrams and in terms of Fomin
growth diagrams. Our analysis uses the fact that $r$-fans of Dyck paths and
vacillating tableaux can be viewed as highest weight elements of weight zero in
crystals of type $B_r$ and $C_r$, respectively, which in turn can be analyzed
using virtual crystals. On the level of Fomin growth diagrams, the
virtualization process corresponds to the Roby-Krattenthaler blow up
construction. One of the motivations for finding rotation invariant
diagrammatic bases such as chord diagrams is the cyclic sieving phenomenon.
Indeed, we give a cyclic sieving phenomenon on $r$-fans of Dyck paths and
vacillating tableaux using the promotion action.

Combinatorics;Representation Theory