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In this paper, we study the combinatorial set of RNA secondary structures oflength $n$ with $m$ base-pairs. For a compact representation, we encode an RNAsecondary structure by the corresponding Motzkin word. For this combinatorialset, we construct an AND/OR tree structure, find a bijection between thecombinatorial set and the set of variants of the AND/OR tree, and developalgorithms for ranking and unranking the variants of the AND/OR tree. Thedeveloped ranking and unranking algorithms have polynomial time complexity$O(m^2 (n - m))$ for $m < n - 2 m$ and $O(m (n - m)^2)$ for $m > n - 2 m$. Incontrast to the existing algorithms, the new algorithms do not requirepreprocessing steps and have better time complexity.

Algorithms for ranking and unranking the combinatorial set of RNA secondary structures