Geodesic
A new bijection between RNA secondary structures and plane trees and its consequences
The electronic journal of combinatorics, Tome 26 (2019) no. 4
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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.
DOI : 10.37236/8540
Classification : 05C05, 05A19, 05A15, 05C10 
@article{10_37236_8540,
     author = {Ricky Xiaofeng Chen},
     title = {A new bijection between {RNA} secondary structures and plane trees and its consequences},
     journal = {The electronic journal of combinatorics},
     year = {2019},
     volume = {26},
     number = {4},
     doi = {10.37236/8540},
     zbl = {1431.05039},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/8540/}
}
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Ricky Xiaofeng Chen. A new bijection between RNA secondary structures and plane trees and its consequences. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/8540

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