In mathematics, a symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains...

polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving...

algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational...

polynomial expression in complete homogeneous symmetric polynomials. The complete homogeneous symmetric polynomial of degree k in n variables X1, ..., Xn, written...

order (i.e., the number of elements) of the symmetric group Sn is n!. A symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that...

terms: it is an alternating polynomial, not a symmetric polynomial. The defining property of the Vandermonde polynomial is that it is alternating in...

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the...

Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle...

thus: the product of two symmetric polynomials is symmetric, the product of a symmetric polynomial and an alternating polynomial is alternating, and the...

in elementary symmetric polynomials of the above variables, for every i, and in the variables yi. Each of the latter symmetric polynomials is a rational...

\ldots ,a_{k-j+1}).} The elementary symmetric polynomial e n {\displaystyle e_{n}} and the power sum symmetric polynomial p n {\displaystyle p_{n}} can be...

of the symmetric polynomials. However, its square ⁠ r 2 2 = ( α − β ) 2 {\displaystyle \textstyle r_{2}^{2}=(\alpha -\beta )^{2}} ⁠ is symmetric in the...

types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one...

symmetric polynomials. In other words, thinking of ai as formal variables, ck "are" σk. A basic fact on symmetric polynomials is that any symmetric polynomial...

polynomials and Vieta's formulas by noting that this expression is a symmetric polynomial in the roots of A. The discriminant of a linear polynomial (degree...

cube, has a general solution. The power sum symmetric polynomial is a building block for symmetric polynomials. The sum of the reciprocals of all perfect...

{1}{n^{2}}}={\frac {\pi ^{2}}{6}}.} Using formulae obtained from elementary symmetric polynomials, this same approach can be used to enumerate formulae for the even-indexed...

indeterminates. Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built...

In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987...

originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For...

once). The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots. Vieta's system (*) can be solved by Newton's method...

the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley (1984) in his study of the symmetric group of permutations...

integers, the a-mean can be equivalently defined via the monomial symmetric polynomial m a ( x 1 , … , x n ) {\displaystyle m_{a}(x_{1},\dots ,x_{n})} as...

Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions...

the symmetric group S 5 {\displaystyle S_{5}} is not solvable, and that there are polynomials with symmetric Galois groups. For n > 4, the symmetric group...

this polynomial may be expanded in a polynomial in s whose coefficients are symmetric polynomials in the xi. By the fundamental theorem of symmetric polynomials...

Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients...

the rank and symmetric rank of a symmetric tensor may differ. Antisymmetric tensor Ricci calculus Schur polynomial Symmetric polynomial Transpose Young...

and only if the eigenvalues of its symmetric part are positive. Symmetric polynomial Elementary symmetric polynomial Newton's identities Invariant theory...