Geodesic
Bijective counting of tree-rooted maps and shuffles of parenthesis systems
The electronic journal of combinatorics, Tome 14 (2007)
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The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is ${\cal C}_{n} {\cal C}_{n+1}$ where ${\cal C}_{n}={1\over n+1}{2n \choose n}$ is the $n^{th}$ Catalan number. We present a (long awaited) simple bijection which explains this result. Then, we prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
DOI : 10.37236/928
Classification : 05A15, 05C30
Mots-clés : Catalan number 
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     author = {Olivier Bernardi},
     title = {Bijective counting of tree-rooted maps and shuffles of parenthesis systems},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/928},
     zbl = {1115.05002},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/928/}
}
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Olivier Bernardi. Bijective counting of tree-rooted maps and shuffles of parenthesis systems. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/928

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