A generalization of the Ramanujan polynomials and plane trees

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 byIn 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, forests of 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.

Derivatives of the tree function

“…Various similar results have been obtained in a rather analytical perspective [14,15], and a probability law related with W (z) as a Bernstein function is studied in [18]. In fact, the polynomials P n are closely related with another sequence that appeared first in Ramanujan's notebook (see [1] and [19]), and inspired quite a few combinatorial works [3,5,11,12,17,24]. The goal of this article is to introduce Greg trees in this context.…”

“…The notion of improper edges in Cayley trees was introduced by Shor [21], and then used by Chen and Guo [3], Guo and Zeng [11], Zeng [24].…”

Arboretum for a generalisation of Ramanujan polynomials

“…Following this, Guo and Zeng study in [8] the following polynomial sequence, credited to Chapoton, as a generalization of the previous sequence…”

“…Study of the Ramanujan's sequence (1) has led to findings regarding refinements to Cayley's formula (see [1,8,14]). This formula gives us that the number of rooted labelled trees with n vertices is n n−1 .…”

A recursive algorithm for trees and forests

“…Aigner and Ziegler's book [1] collected four different proofs of Cayley's formula. We refer the reader to [5,6,8,13,15,20] for several recent results on the enumeration of trees. The goal of this paper is to establish simple linear recurrences between certain forests with roots 1, .…”