TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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The signless first-kind Stirling numbers satisfy their recurrence and expand the rising factorial

Statement

For every $n \geq 1$ and every $k \geq 1$ ,

$c (n, k) = c (n - 1, k - 1) + (n - 1) c (n - 1, k) .$

Moreover, for every $n \in N$ ,

$x^{\overline{n}} = \sum_{k = 0}^{n} c (n, k) x^{k} .$

Facts & Assumptions

Given: The cycle decomposition conventions of Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type and the rising factorial of The rising factorial.

Proof

technique · direct

1.1

To build a permutation of $[n]$ with exactly $k$ cycles, start from a permutation of $[n - 1]$ . Either $n$ forms a new one-cycle, which contributes $c (n - 1, k - 1)$ possibilities, or else $n$ is inserted into one of the $n - 1$ cyclic slots of a permutation with $k$ cycles, which contributes $(n - 1) c (n - 1, k)$ . These two constructions are disjoint and exhaustive, so the recurrence follows.

given

2.1

Let $P_{n} (x) : = \sum_{k = 0}^{n} c (n, k) x^{k}$ . Step 1.1 gives $P_{n} (x) = x P_{n - 1} (x) + (n - 1) P_{n - 1} (x) = (x + n - 1) P_{n - 1} (x)$ , and also $P_{0} (x) = 1$ .

step 1.1algebra

3.1

The rising factorial satisfies the same recursion: $x^{\overline{0}} = 1$ and $x^{\overline{n}} = (x + n - 1) x^{\overline{n - 1}}$ by The rising factorial. Therefore $P_{n} (x) = x^{\overline{n}}$ for all $n$ by induction on $n$ .

step 2.1given∎

Depends on

Used by

The signed first-kind and second-kind Stirling numbers are inverse transition matrices Theorem

Dependency tree · two levels

7 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 (standard reference, not scraped)
Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.5 (standard reference, not scraped)