DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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