CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Compositions of $n$ into $k$ positive parts are counted by $(\binom{n - 1}{k - 1})$

Statement

Let $n, k \in N$ with $n \geq 1$ and $k \geq 1$ . Then the number of compositions of $n$ into exactly $k$ positive parts is

$(\binom{n - 1}{k - 1}) .$

When $k > n$ , there are no such compositions.

Facts & Assumptions

Given: Naturals $n \geq 1$ and $k \geq 1$ .

[L1]

The previous corollary identifies a composition as a finite sequence of positive integers (Positive-integer compositions have generating function $x / (1 - 2 x)$ ).

[L2]

For $m \geq 1$ , the number of weak compositions of $N$ into $m$ parts is $(\binom{N + m - 1}{m - 1})$ (For $m \geq 1$ the number of weak compositions of $n$ into $m$ parts is $(\binom{n + m - 1}{m - 1})$ , and the number of compositions is $(\binom{n - 1}{m - 1})$ for $n \geq 1$ ).

Proof

technique · direct

1.1

A composition $(c_{1}, \dots, c_{k})$ of $n$ into $k$ positive parts determines a weak composition $(c_{1} - 1, \dots, c_{k} - 1)$ of $n - k$ into $k$ parts, and conversely adding $1$ to every part of a weak composition of $n - k$ into $k$ parts recovers a composition of $n$ into $k$ parts. If $k > n$ , no such composition exists, because $c_{1} + \dots + c_{k} \geq 1 + \dots + 1 = k > n$ .

L1constructalgebra

2.1

When $k \leq n$ , step 1.1 and [L2] give $(\binom{(n - k) + k - 1}{k - 1}) = (\binom{n - 1}{k - 1})$ compositions. Together with the empty case from step 1.1, this proves the claim.

step 1.1L2algebra∎

Depends on

Used by

The compositions of 4 from SEQ_≥ 1(SEQ_≥ 1(Z)) Example

Dependency tree · two levels

26 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

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)
Stephen Melczer, An Invitation to Enumeration, Chapter 5: Combinatorial Constructions (standard reference, not scraped)

Compositions of nn into kk positive parts are counted by (n−1k−1)(k−1n−1​)