# nLab elementary symmetric polynomial

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Combinatorics

combinatorics

enumerative combinatorics

graph theory

rewriting

### Polytopes

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category: combinatorics

# Contents

## Definition

The elementary symmetric polynomial on $n$ variables $\{X_i\}$ of degree $k \leq n$ is the polynomial

$\sigma_k(X_1, \cdots, X_n) \coloneqq \sum_{1 \leq i_1 \lneq \cdots \lneq i_k \leq n} X_{i_1} \cdots X_{i_k} \,.$

Equivalently these are the degree-$k$ summands in the polynomial

$(1+X_1)(1+X_2) \cdots (1+X_n)$

These polynomials form a basis for the Lambda-ring of symmetric functions.

## References

Last revised on September 3, 2021 at 04:31:11. See the history of this page for a list of all contributions to it.