We study Muirhead-type generalizations of families of inequalities due to Newton, Maclaurin and others. Each family is defined in terms of a commonly used basis of the ring of symmetric functions in n variables. Inequalities corresponding to elementary symmetric functions and power sum symmetric functions are characterized by the same simple poset which generalizes the majorization order. Some ...

An association scheme is amorphous if it has as many fusion schemes as possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V. Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of combinatorial objects, pages 39–49. VNIISI, Moscow, Institute for System Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A. Munemasa and M. Yam...

For a given pair of s-dimensional real Laurent polynomials (~a(z),~b(z)), which has a certain type of symmetry and satisfies the dual condition~b(z) T ~a(z) = 1, an s× s Laurent polynomial matrix A(z) (together with its inverse A−1(z)) is called a symmetric Laurent polynomial matrix extension of the dual pair (~a(z),~b(z)) if A(z) has similar symmetry, the inverse A−1(Z) also is a Laurent polyn...

The growth function of a graph with respect to a vertex is near polynomial if there exists a polynomial bounding it above for infinitely many positive integers. In the paper vertex-symmetric undirected graphs and vertex-symmetric directed graphs with coinciding inand out-degrees are described in the case their growth functions are near polynomial.

Abstract. The number of real roots of a system of polynomial equations fitting inside a given box can be counted using a vector symmetric polynomial introduced by P. Milne, the volume function. We provide the expansion of Milne’s volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear in the expansion, the squaref...

1. Macdonald polynomials Macdonald polynomials P λ (x; q, t) are orthogonal symmetric polynomials which are the natural multivariable generalisation of the continuous q-ultraspherical polyno-mials C n (x; β|q) [2] which, in their turn, constitute an important class of hyper-geometric orthogonal polynomials in one variable. Polynomials C n (x; β|q) can be obtained from the general Askey-Wilson p...

Abstract. Through an algebraic method using the Dunkl–Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the AN−1and BN Calogero models with bosonic, fermionic and distinguishable particles are investigated. The Rodrigues formulas of column type that algebraically generate the monic nonsymmetric multivariable Hermite and Laguerre polynomials corresponding to...

Abstract Let R(X) = Q[x1,x2 xn] be the ring of polynomials in the variables X = {x1, X2,..., xn} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a a e Sn, we let g a ( X ) = Yai>ai+1 (xa1,xa2 ... xai). In the late 1970s I. Cessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result ...

We derive absolute perturbation bounds for the coefficients of the characteristic polynomial of a n × n complex matrix. The bounds consist of elementary symmetric functions of singular values, and suggest that coefficients of normal matrices are better conditioned with regard to absolute perturbations than those of general matrices. When the matrix is Hermitian positivedefinite, the bounds can ...