# SymmetricPolynomial

SymmetricPolynomial[k,{x1,,xn}]

gives the k elementary symmetric polynomial in the variables x1,,xn.

# Details

• A symmetric polynomial of n variables {x1,,xn} is invariant under any permutation of its variables. The k elementary symmetric polynomial is the sum of all square-free monomials of degree k.
• The degree k must satisfy 0kn.
• The elementary symmetric polynomials form a basis for the symmetric polynomials.
• Expressing a general symmetric polynomial in terms of elementary symmetric polynomials is accomplished by using SymmetricReduction.

# Examples

open allclose all

## Basic Examples(1)

The elementary symmetric polynomial of degree 3 in variables x1,x2,x3,x4:

## Scope(1)

The zeroth elementary symmetric polynomial is defined to be 1:

## Applications(1)

The 2×3 matrices with entries 0 or 1:

Select matrices whose column sums are 1,1,1 and whose row sums are 2,1:

You can also count how many such matrices there are by using SymmetricPolynomial. The generating function of 2×3 matrices whose row sums are 2,1 is given by:

The coefficient of x11x21x31 counts how many of these matrices have column sums 1,1,1:

## Properties & Relations(5)

The k elementary symmetric polynomial is the sum of all monomials constructed from k-subsets of the variables:

The generating function for the symmetric polynomials in variables is given by :

Check:

The monic polynomial with roots has coefficients that are elementary symmetric polynomials of the :

The elementary symmetric polynomials ek=SymmetricPolynomial[k,{x1,,xn}] are related to the power sum polynomials through the NewtonGirard formulas [MathWorld]. Generate all the NewtonGirard formulas for :

Verify them:

The elementary symmetric polynomials can be defined in terms of the generalized Bell polynomial BellY:

Verify for the case of five variables:

## Neat Examples(1)

Find integers such that the roots of are :

Check:

Wolfram Research (2007), SymmetricPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricPolynomial.html.

#### Text

Wolfram Research (2007), SymmetricPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricPolynomial.html.

#### CMS

Wolfram Language. 2007. "SymmetricPolynomial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SymmetricPolynomial.html.

#### APA

Wolfram Language. (2007). SymmetricPolynomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SymmetricPolynomial.html

#### BibTeX

@misc{reference.wolfram_2024_symmetricpolynomial, author="Wolfram Research", title="{SymmetricPolynomial}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetricPolynomial.html}", note=[Accessed: 22-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_symmetricpolynomial, organization={Wolfram Research}, title={SymmetricPolynomial}, year={2007}, url={https://reference.wolfram.com/language/ref/SymmetricPolynomial.html}, note=[Accessed: 22-July-2024 ]}