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【期刊论文】Bijections behind the Ramanujan Polynomials
陈永川, William Y. C. Chen, Victor J. W. Guo
Advances in Applied Mathematics 27, 336-356 (2001),-0001,():
-1年11月30日
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.
Ramanujan polynomials, bijection, rooted tree, Improper edge.,
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【期刊论文】The Pessimistic Search and the Straightening Involution for Trees
陈永川, WILLIAM Y. C. CHEN
Europ. J. Combinatorics (1998) 19, 553-558,-0001,():
-1年11月30日
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.
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【期刊论文】The Combinatorics of a Class of Representation Functions
陈永川, William Y. C. Chen and James D. Louck
Advances in Mathematics 140, 207-236 (1998),-0001,():
-1年11月30日
, 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.
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【期刊论文】The factorial Schur function
陈永川, William Y.C. Chen a), James D.Louck
J. Math. Phys. 34 (9), September 1993,-0001,():
-1年11月30日
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.
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