False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-02 (claude-opus-5)

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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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FALSE: the number of weak compositions of $n$ into $m$ parts is $\binom{n+m-1}{m-1}$ for every $m \in \mathbb{N}$

Statement

FALSE. The statement

$\big\lvert\mathcal{W}(n,m)\big\rvert = \dbinom{n+m-1}{\,m-1\,}$

for every $n \in \mathbb{N}$ and every $m \in \mathbb{N}$ , that is, For $m \ge 1$ 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 $n \ge 1$ with its hypothesis $m \ge 1$ deleted.

This is a false statement of an unusual kind: at $m = 0$ the expression on the right is not even well formed under the reading a reader would intend, and under the only reading available in this library it is well formed and gives the wrong number.

Facts & Assumptions

Given: The sets $\mathcal{W}(n,m)$ of Compositions and weak compositions of a natural number into a fixed number of parts; binomial coefficients defined for natural arguments only (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ ); and the truncated difference of Finite sums and finite products of natural numbers, $\sum_{k<n} a_k$ and $\prod_{k<n} a_k$ in $\mathbb{N}$ , under which $n - m$ is $0$ whenever $n < m$ .

[L1]

$\lvert\mathcal{W}(0,0)\rvert = 1$ and $\lvert\mathcal{W}(n,0)\rvert = 0$ for $n \ge 1$ (Compositions and weak compositions of a natural number into a fixed number of parts, The cardinality $\lvert A\rvert$ of a finite set).

[L2]

$\binom{N}{0} = 1$ and $\binom{N}{N} = 1$ for every natural $N$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ ).

[L4]

$\binom{n}{k}$ is defined only for $n, k \in \mathbb{N}$ , and $-1$ is not a natural number (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ , The natural numbers $\mathbb{N}$ (von Neumann), Order on the natural numbers).

[L5]

$0 \ne 1$ (Trichotomy of the order on $\mathbb{N}$ , The natural numbers $\mathbb{N}$ (von Neumann)).

Refutation

technique · direct

1.1

Fix $m = 0$ and $n = 1$ . The true count is $\lvert\mathcal{W}(1,0)\rvert = 0$ by [L1]: a weak composition of $1$ into $0$ parts would be a function $0 \to \mathbb{N}$ , and the only such function is the empty function, whose sum is the empty sum $0 \ne 1$ .

givenL1L5

2.1

The formula gives $1$ . With the truncated difference, $n+m-1 = 1+0-1 = 0$ and $m-1 = 0-1 = 0$ , so the right-hand side reads $\binom{0}{0} = 1$ by [L2]. Since $1 \ne 0$ by [L5], the displayed statement is false at $(n,m) = (1,0)$ .

step 1.1L2L5

3.1

Under the other reading the expression is not defined at all. If $m-1$ is meant as an integer, it is $-1$ at $m = 0$ , and $-1$ is not a natural number, so $\binom{n-1}{-1}$ names nothing: The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ defines $\binom{N}{k}$ for natural $N$ and $k$ only. So the statement is either false or ill formed, and in neither reading is it true.

step 2.1L4∎

Remarks

The formula is correct for every $m \ge 1$ , which is what For $m \ge 1$ 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 $n \ge 1$ asserts: $\lvert\mathcal{W}(n,m)\rvert = \binom{n+m-1}{m-1}$ under that hypothesis. The two edges are worth seeing. At $m = 1$ it reads $\binom{n}{0} = 1$ , matching the unique weak composition $(n)$ . At $n = 0$ it reads $\binom{m-1}{m-1} = 1$ , matching the unique weak composition all of whose parts are $0$ . Neither of these is the failing case; the failure is confined to $m = 0$ .
A false statement whose falsity is ill-formedness is worth stating in exactly those terms. What For $m \ge 1$ 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 $n \ge 1$ asserts is a statement about $m \ge 1$ ; the object $\binom{n+m-1}{m-1}$ simply does not exist at $m = 0$ unless one adopts a truncation convention, and adopting one makes the value wrong rather than absent.
The companion true values are recorded in Compositions and weak compositions of a natural number into a fixed number of parts: at $m = 0$ there is exactly one weak composition of $0$ and none of any $n \ge 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 76 results over 30 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Stars and bars (combinatorics) (Wikipedia) (standard reference, not scraped)
Composition (combinatorics) (Wikipedia) (standard reference, not scraped)
Binomial coefficient (Wikipedia) (standard reference, not scraped)

FALSE: the number of weak compositions of nnn into mmm parts is (n+m−1m−1)\binom{n+m-1}{m-1}(m−1n+m−1​) for every m∈Nm \in \mathbb{N}m∈N