Old and young leaves on plane trees

Authors:
William Y. C. Chen;Emeric Deutsch;Sergi Elizalde
Affiliations:
Center for Combinatorics, LPMC, Nankai University, Tianjin, PR China;Polytechnic University, Brooklyn, NY;Massachusetts Institute of Technology, Cambridge, MA and Mathematical Sciences Research Institute, Berkeley, CA
Venue:
European Journal of Combinatorics
Year:
2006

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Abstract

A leaf of a plane tree is called an old leaf if it is the leftmost child of its parent, and it is called a young leaf otherwise. In this paper we enumerate plane trees with given numbers of old leaves and young leaves. The formula is obtained combinatorially via two bijections between plane trees and 2-Motzkin paths which map young leaves to red horizontal steps, and old leaves to up steps. We derive some implications for the enumeration of restricted permutations with respect to certain statistics such as pairs of consecutive deficiencies, double descents, and ascending runs. Finally, our main bijection is applied to obtain refinements of two identities of Coker, involving refined Narayana numbers and the Catalan numbers.