BUILT-IN MATHEMATICA SYMBOL

SymmetricPolynomial

gives the k

elementary symmetric polynomial in the variables

.

MORE INFORMATION

A symmetric polynomial of n variables 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 .

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

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Basic Examples (1)

The elementary symmetric polynomial of degree 3 in variables

:

The elementary symmetric polynomial of degree 3 in variables

:

In[1]:=

Out[1]=

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

and whose row sums are

:

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

is given by:

The coefficient of

counts how many of these matrices have column sums

:

Properties & Relations (4)

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

are related to the power sum polynomials

through the Newton-Girard identities:

For example, with

:

Neat Examples (1)

Find integers

such that the roots of

are

:

Check: