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1.
Let R(X) = Q[x 1, x 2, ..., x n] be the ring of polynomials in the variables X = {x 1, x 2, ..., x n} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a S n, we let g In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x 1, x 2, ..., x n} and Y = {y 1, y 2, ..., y n}. The diagonal action of S n on polynomial P(X, Y) is defined as Let R (X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R *(X, Y) denote the quotient of R (X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for R *(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and R *(X, Y) in terms of their respective bases.  相似文献   

2.
Stanley (Advances in Math. 111, 1995, 166–194) associated with a graph G a symmetric function X G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about , as well as new ones that cannot be interpreted on the level of the chromatic polynomial. Unfortunately, X G does not satisfy a Deletion-Contraction Law which makes it difficult to apply the useful technique of induction. We introduce a symmetric function Y G in noncommuting variables which does have such a law and specializes to X G when the variables are allowed to commute. This permits us to further generalize some of Stanley's theorems and prove them in a uniform and straightforward manner. Furthermore, we make some progress on the (3 + 1)-free Conjecture of Stanley and Stembridge (J. Combin Theory (A) J. 62, 1993, 261–279).  相似文献   

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