DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Monomial symmetric polynomials indexed by partitions

Definition

A partition of length at most $n$ is a tuple $λ = (λ_{1}, \dots, λ_{n})$ of natural numbers with $λ_{1} \geq \dots \geq λ_{n}$ . Its monomial symmetric polynomial is

$m_{λ} (x_{1}, \dots, x_{n}) : = \sum_{a \in Orb (λ)} x_{1}^{a_{1}} \dots x_{n}^{a_{n}},$

where $Orb (λ)$ is the set of distinct tuples obtained by permuting the coordinates of $λ$ . Thus repeated monomials are counted once, not with their stabilizer multiplicity. The sum is symmetric by construction.

Depends on

Symmetric polynomials as the invariants of variable permutations

Used by

Monomial symmetric polynomials form an R-basis of the symmetric-polynomial ring Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 8 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)