Enumerative combinatorics
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Parking functions, valet functions and priority queues
Discrete Mathematics
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We investigate a particular symmetry in labeled trees first discovered by Gessel, which can be stated as follows: In the set of rooted labeled trees on n + 1 vertices rooted at the smallest vertex, the number of trees with a descents and b + 1 leaves equals the number of trees with b descents and a + 1 leaves. We present two new ways to prove the symmetry resulting from decompositions of trees, which lead to three different bijections from trees to trees in which leaves and descents are swapped. We also interpret the symmetry in terms of parking functions: the number of parking functions on [n] with a descents and b unfavorable spaces (defined in this paper) equals the number of parking functions on [n] with b descents and a unfavorable spaces. We conclude with a generalization of these results to binary trees.