An embedding of the skein action on set partitions into the skein action on matchings

Rhoades defined a skein action of the symmetric group on the linear span of noncrossing set partitions which generalized an action of the symmetric group on the linear span of matchings. The en-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' en-module can be embedded into the en-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.