Bijections behind the Ramanujan Polynomials
William Y. C. Chen and Victor. J. W. Guo
The Ramanujan polynomials were introduced by Ramanujan in his
study of power series inversions.
In an approach to the Cayley formula on the number of trees, Shor discovers
a refined recurrence relation in terms of the number of improper edges,
without realizing the connection to the Ramanujan polynomials. On
the other hand, Dumont and Ramamonjisoa independently take the
grammatical approach to a sequence associated with the Ramanujan
polynomials and have reached the same conclusion as Shor's.
It was a
coincidence for Zeng to realize that the Shor polynomials turn out
to be the Ramanujan polynomials through an explicit substitution
of parameters. On the other side of the story,
Shor also discovers a recursion of Ramanujan
polynomials which is equivalent to the Berndt-Evans-Wilson
recursion under the substitution of Zeng, and asks for a
combinatorial interpretation. The objective of this paper is to
present a bijection for the Shor recursion, or and
Berndt-Evans-Wilson recursion, answering the question of Shor.
Such a bijection also leads to a combinatorial interpretation of
the recurrence relation originally given by Ramanujan.