Before revision: We show that the notion of induction introduced by Cassaigne, Ferenczi and Zamboni for trees of relations arising in the context of interval exchange relations can be generalised to the case of an arbitrary number of possible edge labels. We prove that the equivalence classes of its transitive closure can still be characterised via a circular order on the trees of relations in this case. We compute the cardinalities of these equivalence classes and show that the sequence of cardinalities, for a fixed number of possible edge labels, is a convolution of a Fuss-Catalan sequence. As in the original case, the equivalence classes are in bijection with a set of pseudoknot-free secondary structures arising from the study of RNA; we show that a natural subset of this set is in bijection with a set of m-clusters (in the cluster algebra sense).
After revision: We develop and study the structure of combinatorial objects arising from interval exchange transformations. We classify and enumerate these objects and further show that a natural subset of this set is in bijection with a set of m-clusters (in the cluster algebra sense).