Combinatorics on Plane Trees, Motivated by RNA Secondary Structure Configurations

Motivated by the base pairing of RNA sequences, and as part of our collaboration with A. Condon and H. H. Hoos, we defined the local move operation on two unobstructed edges in a plane tree given here. We now consider the graph Gn induced by this operation on Tn, the set of plane trees with n edges. We provide a series of results showing that Gn is a connected, n-partite graph of diameter n− 1 with disjoint sets whose cardinalities are enumerated by the Narayana numbers. Our partition of Gn depends on orienting the edges in a plane tree, and differs from the known Narayana decomposition of ordered trees according to the number of leaves by N. Dershowitz and S. Zaks. We then consider the partial ordering induced by an antisymmetric restriction of our local move operation. After a theorem characterizing the new relation, we prove that the poset (Tn, ) is a lattice, which is complemented but not distributive. Finally, in a joint result with S. Fomin, we show that (Tn, ) is isomorphic to the lattice of noncrossing partitions.