In mathematics, the term "symmetric function" can mean two different concepts.

Since is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!

This final point applies in particular to the family (h) of complete homogeneous symmetric functions.

Timing attacks can potentially be used against other cryptosystems, including symmetric functions.

Expanding out yields the elementary symmetric functions of the :

The following are fundamental examples of symmetric functions.

Define where is a non-negative symmetric function in and that can be chosen by the user.

This phenomenon can be understood in the setting of the ring of symmetric functions.

The obvious fact that explains the symmetry between elementary and complete homogeneous symmetric functions.

In the same way, all the Elementary symmetric functions are Schur-concave, when .

It is a symmetric function of its arguments:

Each U-statistic is necessarily a symmetric function.

Hence the product of i and f has to be a totally symmetric function, too, otherwise the matrix element would vanish.

A symmetric matrix, seen as a symmetric function of the row- and column number, is an example.

In turn this implies that the Schur function of a partition is a symmetric function.

This is overcome by treating as a fully symmetric function of five arguments, two of which happen to have the same value .

The study of symmetric functions is based on that of symmetric polynomials.

The general solution for arbitrary n can be expressed in terms of symmetric functions of and .

A symmetric function inserts at the end:

The left hand sides of Vieta's formulas are the elementary symmetric functions of the roots.

A ring of this kind plays a role in constructing the ring of symmetric functions.

The second definition of the ring of symmetric functions implies the following fundamental principle:

His 1979 book Symmetric Functions and Hall Polynomials has become a classic.

This ring generalizes the ring of symmetric functions.

Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions.