DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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.

The $q$ -integer, $q$ -factorial and $q$ -multinomial coefficients

Definition

For $n \in N$ , the $q$ -integer is

$[n]_{q} : = 1 + q + \dots + q^{n - 1},$

with $[0]_{q} : = 0$ .

The $q$ -factorial is

$[n]_{q}! : = \prod_{i = 1}^{n} [i]_{q},$

so $[0]_{q}! : = 1$ .

If $k_{1}, \dots, k_{r} \in N$ satisfy $k_{1} + \dots + k_{r} = n$ , the $q$ -multinomial coefficient is

${(\binom{n}{k_{1}, \dots, k_{r}})}_{q} : = \frac{[n]_{q}!}{[k_{1}]_{q}! \dots [k_{r}]_{q}!} .$

These are polynomials in $Z [q]$ once the integrality theorems of the page are proved; at this point they are formal expressions to be identified by their counting roles.

Depends on

The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$

Used by

The inversion generating function of Sₙ is [n]_q! Theorem

Dependency tree · two levels

15 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

Andrew Lin, MIT 18.212 Algebraic Combinatorics, Lecture 9 (standard reference, not scraped)
Felix Gotti, MIT 18.211 Combinatorial Analysis, Lecture 5 (standard reference, not scraped)

The qq-integer, qq-factorial and qq-multinomial coefficients