Built-in Mathematica Symbol

SymmetricPolynomial

SymmetricPolynomial[k, {x₁, ..., x_n}]
gives the k

elementary symmetric polynomial in the variables x₁, ..., x_n.

MORE INFORMATION

A symmetric polynomial of n variables {x₁, ..., x_n} 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 0≤k≤n.

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 x₁, x₂, x₃, x₄:

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 1, 1, 1 and whose row sums are 2, 1:

We 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 (4)

The generating function for the symmetric polynomials in