k-foldability of words

We extend results regarding a combinatorial model introduced by Black et al.(2017) which generalizes the folding of the RNA molecule in biology. Consider a word on alphabet {A 1 ,A¯ 1 ,…,A m ,A¯ m } in which A¯ i is called the complement of A i . A word w is foldable if can be wrapped around a rooted plane tree T, starting at the root and working counterclockwise such that one letter labels each half edge and the two letters labeling the same edge are complements. The tree T is called w-valid. We define a bijection between edge-colored plane trees and words folded onto trees. This bijection is used to characterize and enumerate words for which there is only one valid tree. We follow up with a characterization of words for which there exist exactly two valid trees. In addition, we examine the set R(n,m) consisting of all integers k for which there exists a word with the alphabet {A 1 ,A¯ 1 ,…,A m ,A¯ m } of length 2n with exactly k valid trees. Black, Drellich, and Tymoczko showed that for the nth Catalan number C n , {C n ,C n−1 }⊂R(n,1) but k⁄∈R(n,1) for C n−1 <k<C n . We describe a superset of R(n,1) in terms of the Catalan numbers by which we establish more missing intervals. We also prove R(n,1) contains all non-negative integers less than n+1.