We introduce the idea of pessimistic search on a rooted tree, and develop the straightening involution to relate the inversion polynomial evaluated at q D −1 to the number of even rooted trees. We obtain a differential equation for the inversion polynomial of cyclic trees evaluated at q D −1, a problem proposed by Gessel, Sagan and Yeh. Some brief discussions about relevant topics are also presented.

Words on two letters, or their equivalent representation by ɑ sequences, label the branches of the inverse graph of the nth iterate of the parabolic map pζ (x)=ζx(2-x) of the real line. The abstract properties of words control the evolution of this graph in the content parameter. In particular, properties of words (ɑ sequences) control the process of creation and bifurcation of fixed points. The subset of lexical words of length n-1 or the corresponding set of lexical ɑ sequences of degree D=n-1 are key entities in this description, as are the divisor set of lexical words of degree D such that 1+D divides n. The parity, even or odd, of the length of the lexical sequences in the divisor set controls the motion, from left to right or right to left, of the central point (1, x(ζ)) of the inverse graph through the midpoint (1, 1), as the content parameter increase. It is proved in this paper that adjacent sequences in the ordered divisor set alternate in the parity of their lengths, this then corresponding to an oscillatory motion of the central point back and forth through the central point. The abstract parity property of words thus corresponds to an important property of the inverse graph in its evolution in the content parameter.

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. 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 the 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.

We present an involution for a classical identity on the alternate sum of the Gauss coefficients in terms of the traditional Ferrers diagram. It turns out that the refinement of our involution with restrictions on the height of Ferrers diagram implies a generalization of the Gauss identity, which is a terminating form of the q-Kummer identity. Furthermore, we extend the Gauss identity to the pth root of unity.