Abstract
In this paper, we first present a new bijection between RNA secondary structures and plane trees. Combined with the Schmitt-Waterman bijection between these objects, we then obtain a bijection on plane trees that relates the horizontal fiber decomposition associated to internal vertices to the degrees of odd-level vertices while the vertical path decomposition associated to leaves is related to the degrees of even-level vertices. To the best of our knowledge, only the former relation (i.e., horizontal vs odd-level) due to Deutsch is known. As a consequence, we obtain enumeration results for various classes of plane trees, e.g., refining the Narayana numbers and the enumeration involving young leaves due to Chen, Deutsch and Elizalde, and counting a newly introduced `vertical' version of $k$-ary trees. The enumeration results can be also formulated in terms of RNA secondary structures with certain parameterized features, which might have some biological significance.
Highlights
Ribonucleic acid (RNA) plays an important role in various biological processes within cells, ranging from catalytic activity to gene expression
Enumeration of the electronic journal of combinatorics 26(4) (2019), #P4.48 the number of secondary structures over a sequence of length n that have k base pairs has been done in Schmitt and Waterman [6] by establishing a bijection between secondary structures and plane trees
We present a new bijection between RNA secondary structures and plane trees
Summary
Ribonucleic acid (RNA) plays an important role in various biological processes within cells, ranging from catalytic activity to gene expression. Enumeration of the electronic journal of combinatorics 26(4) (2019), #P4.48 the number of secondary structures over a sequence of length n that have k base pairs has been done in Schmitt and Waterman [6] by establishing a bijection between secondary structures and plane trees. Through our new bijection φ on plane trees, can we show the correspondence between horizontal fibers and odd-level vertices established by Deutsch [4], but we can show that the vertical paths associated to leaves correspond to even-level vertices at the same time. We are interested in their vertical duals, i.e., plane trees where any vertical path associated to a leaf somehow has a size k We show that these ‘vertical’ k-ary trees are counted by numbers very similar to the generalized Catalan numbers. We refine some results obtained in Chen, Deutsch and Elizalde [2]
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