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.

, 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 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.

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.

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.