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1.
E.E. Allen 《Journal of Algebraic Combinatorics》1994,3(1):5-16
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