Trees are combinatorial structures that arise naturally in diverse applications. They occur in branching decision structures, taxonomy, computer languages, combina-torial optimization, parsing of sentences, and cluster expan-sions of statistical mechanics. Intuitively, a tree is a collection of branches connected at nodes. Formally, it can be defined as a connected graph without cycles. Schroder trees, introduced in this paper, are a class of trees for which the set of subtrees at any vertex is endowed with the structure of ordered parti-tions. An ordered partition is a partition of a set in which the blocks are linearly ordered. Labeled rooted trees and labeled planed trees are both special classes of Schroder trees. The main result gives a bijection between Schroder trees and forests of small trees-namely, rooted trees of height one. Using this bijection, it is easy to encode a Schroder tree by a sequence of integers. Several classical algorithms for trees, including a combinatorial proof of the Lagrange inversion formula, are immediate consequences of this bijection.

In this paper, we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substituion rules. We call such a set of rules a context-free grammar because these rules act like a context-free grammar in the sense of a formal language. Given a context-free grammar, we can associate each formal function with an exponential formal power series. In this way, we obtain grammatical interpretations of addition, multiplication and functional composition of formal power series. A surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation. We apply this method to obtain simple demonstrations of Faa di Bruno's formula, and some identities concerning Bell polynomials, Stirling numbers and symmetric functions. In particular, the Lagrange inversion formula has a simple grammatical representation. From this point of view, one sees that Cayley's formula on labeled trees is equivalent to the Lagrange inversion formula.

The application of the divided difference of a function to the inhomogeneous symmetric functions (factorial Schur functions) of Biedenharn and Louck is shown to head to new relations and simplified proofs of their properties. These results include determinantal definitions and the factorial Jacobi-Trudi identities with extensions to skew versions. Similar properties of a second class of sym-metric functions depending on an arbitrary parameter, and of importance for generalized hypergeometric functions and series, are shown also to be derivable from the divided difference notion, slightly extended.

We obtain an interpolation formula for symmetric functions and applications to some identities on symmetric functions, including the one obtained by Gustafson and Milne on Schur functions.

, j n, which may be arranged into an n_n matrix array Z=(zij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrix D(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable "group multiplication property" D(XY)=D(X) D(Y) for arbitrary matrices X and Y, even though these matrices may be singular. For Z=U # U(n), these matrices give irreducible unitary representations of U(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to under-standing group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted by LXβ (Z), where ɑ and β are integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equiv-alent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n-j coef-ficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties.

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.