TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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.

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Ordinary powers expand in the falling-factorial basis by the second-kind Stirling numbers

Statement

For all $n, m \in N$ ,

$m^{n} = \sum_{k = 0}^{n} S (n, k) m^{\underline{k}} .$

Facts & Assumptions

Given: Naturals $n$ and $m$ .

Proof

technique · direct

1.1

The left-hand side $m^{n}$ counts all functions $[n] \to [m]$ . Group those functions by the partition of $[n]$ into their nonempty fibres. If the image has size $k$ , then the fibres form a partition of $[n]$ into $k$ blocks, counted by $S (n, k)$ .

given

1.2

Once such a $k$ -block partition is fixed, assigning distinct values of $[m]$ to its blocks is equivalent to choosing an ordered $k$ -tuple of distinct elements of $[m]$ , and there are exactly $m^{\underline{k}}$ such choices by The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ .

given

2.1

Summing over all possible image sizes $k$ counts every function $[n] \to [m]$ exactly once, so $m^{n} = \sum_{k = 0}^{n} S (n, k) m^{\underline{k}}$ .

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

33 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, 18.212 Algebraic Combinatorics, Lecture 11, Theorem 2 (standard reference, not scraped)