DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$

Definition

For $k \geq 1$ , the $k$ -th power-sum symmetric polynomial is

$p_{k} : = x_{1}^{k} + \dots + x_{n}^{k} .$

For $k \geq 0$ , the $k$ -th complete homogeneous symmetric polynomial is

$h_{k} : = \sum_{a_{1} + \dots + a_{n} = k} x_{1}^{a_{1}} \dots x_{n}^{a_{n}} .$

Thus $h_{0} = 1$ , while for $n = 0$ one has $h_{k} = 0$ when $k > 0$ . Both families are fixed by every permutation of the variables.

Depends on

Used by

The generating-series identity E(-t)H(t)=1 Proposition
Newton's identities: k eₖ=∑ᵢ₌₁ᵏ(-1)ⁱ⁻¹eₖ₋ᵢpᵢ Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 43 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

D. Grinberg, An Introduction to Algebraic Combinatorics, Chapter 7, Sections 7.1-7.2 (standard reference, not scraped)

Power sums pkpk​ and complete homogeneous symmetric polynomials hkhk​