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28 March 2024
 
  » arxiv » math.CO/0503300

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Parity Reversing Involutions on Plane Trees and 2-Motzkin Paths
William Y.C. Chen ; Louis W. Shapiro ; Laura L.M. Yang ;
Date 15 Mar 2005
Subject Combinatorics MSC-class: 05A15, 05C30, 05C05 | math.CO
AbstractThe problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schröder paths, and it was recently derived by Coker using the Lagrange inversion formula. An equivalent problem for partitions was independently studied by Klazar. We present three parity reversing involutions, one for unlabelled plane trees, the other for labelled plane trees and one for 2-Motzkin paths which are in one-to-one correspondence with Dyck paths.
Source arXiv, math.CO/0503300
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