A Generalization of the Ramanujan Polynomials and Plane Trees

Details A Generalization of the Ramanujan Polynomials and Plane Trees A Generalization of the Ramanujan Polynomials and Plane Trees

2005-12-12

arxiv.org/abs/math/0512249v2

Adv. Appl. Math., 39 (2007), 96--115

Mathematics

Combinatorics

20 pages, 2 tables, 8 figures, see also http://math.univ-lyon1.fr/~guo

Scientific paper

10.1016/j.aam.2006.01.004

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.

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