A Generalization of the Ramanujan Polynomials and Plane Trees
arXiv:math/0512249 · doi:10.1016/j.aam.2006.01.004
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.
20 pages, 2 tables, 8 figures, see also http://math.univ-lyon1.fr/~guo
References in corpus (1)
Cited by in corpus (7)
- Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees
- A recursive algorithm for trees and forests
- Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs
- Disposition Polynomials and Plane Trees
- Arboretum for a generalization of Ramanujan polynomials
- Derivatives of the tree function
- Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials