Geodesic
Counting forests by descents and leaves
The electronic journal of combinatorics, The Foata Festschrift volume, Tome 3 (1996) no. 2
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A descent of a rooted tree with totally ordered vertices is a vertex that is greater than at least one of its children. A leaf is a vertex with no children. We show that the number of forests of rooted trees on a given vertex set with $i+1$ leaves and $j$ descents is equal to the number with $j+1$ leaves and $i$ descents. We do this by finding a functional equation for the corresponding exponential generating function that shows that it is symmetric.
DOI : 10.37236/1266
Classification : 05C30, 05C05
Mots-clés : counting forests, descent, rooted tree, leaf 
@article{10_37236_1266,
     author = {Ira M. Gessel},
     title = {Counting forests by descents and leaves},
     journal = {The electronic journal of combinatorics},
     year = {1996},
     volume = {3},
     number = {2},
     doi = {10.37236/1266},
     zbl = {0858.05054},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1266/}
}
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Ira M. Gessel. Counting forests by descents and leaves. The electronic journal of combinatorics, The Foata Festschrift volume, Tome 3 (1996) no. 2. doi: 10.37236/1266

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