A Generalization of the Ramanujan Polynomials and Plane Trees
Combinatorics
2011-03-25 v2
Abstract
Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1, Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and answer his question about the combinatorial interpretation of Q_n. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees.
Cite
@article{arxiv.math/0512249,
title = {A Generalization of the Ramanujan Polynomials and Plane Trees},
author = {Victor J. W. Guo and Jiang Zeng},
journal= {arXiv preprint arXiv:math/0512249},
year = {2011}
}
Comments
20 pages, 2 tables, 8 figures, see also http://math.univ-lyon1.fr/~guo