This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)

SymmetricPolynomial

SymmetricPolynomial
gives the k^(th) elementary symmetric polynomial in the variables .
  • A symmetric polynomial of n variables is invariant under any permutation of its variables. The k^(th) 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.
The elementary symmetric polynomial of degree 3 in variables :
The elementary symmetric polynomial of degree 3 in variables :
In[1]:=
Click for copyable input
Out[1]=
The zeroth elementary symmetric polynomial is defined to be 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 :
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 :
Find integers such that the roots of are :
Check:
New in 6