TheoremStatement: Literature-sourcedProof: Literature-sourcedaudited 2026-08-28

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Euler's pentagonal number theorem by Franklin's involution

Statement

In $Z ⟦ x ⟧$ ,

$\prod_{m \geq 1} (1 - x^{m}) = \sum_{r \in Z} (- 1)^{r} x^{r (3 r - 1) / 2} .$

Equivalently,

$\prod_{m \geq 1} (1 - x^{m}) = 1 + \sum_{r \geq 1} (- 1)^{r} (x^{r (3 r - 1) / 2} + x^{r (3 r + 1) / 2}) .$

Facts & Assumptions

Given: for each $n \geq 0$ , let $q_{e} (n)$ and $q_{o} (n)$ denote the numbers of partitions of $n$ into an even, respectively odd, number of distinct parts.

[F1]

A partition into distinct parts is a finite strictly decreasing list of positive integers; $q_{e} (n)$ and $q_{o} (n)$ count the even-length and odd-length such partitions of $n$ (The functions p(n), p_k(n), and the standard restricted partition families).

[F2]

Expanding $\prod_{m \geq 1} (1 - x^{m})$ chooses each part size $m$ either not at all or once, with sign $- 1$ when it is chosen, so the coefficient of $x^{n}$ is $q_{e} (n) - q_{o} (n)$ .

Proof

technique · sign-reversing involution

1.1F1construct

Let $λ = (λ_{1} > \dots > λ_{ℓ})$ be a nonempty partition into distinct parts. Write $s = λ_{ℓ}$ for its smallest part, and let $t$ be the largest index such that $λ_{i} = λ_{1} - i + 1$ for every $1 \leq i \leq t$ . Thus the first $t$ rows form the maximal upper-right staircase. If $s \leq t$ and $λ \neq (2 s - 1, 2 s - 2, \dots, s)$ , define $α (λ)$ by deleting the last part $s$ and adding $1$ to each of the first $s$ parts. If $s > t$ and $λ \neq (2 t, 2 t - 1, \dots, t + 1)$ , define $β (λ)$ by subtracting $1$ from each of the first $t$ parts and adjoining a new last part $t$ .

2.1step 1.1algebra

In the first case of step 1.1, the partition $α (λ)$ is still distinct: the first $s$ parts remain strictly decreasing, the last changed part satisfies $λ_{s} + 1 > λ_{s + 1}$ because those rows were consecutive, and deleting the old last part decreases the number of parts by $1$ . The first $s$ parts of $α (λ)$ are consecutive and its smallest part is larger than $s$ , so $α (λ)$ falls under the second construction with parameter $t = s$ , and $β (α (λ)) = λ$ .

2.2step 1.1algebra

In the second case of step 1.1, the partition $β (λ)$ is still distinct: maximality of $t$ gives $λ_{t} - 1 > λ_{t + 1}$ , while the new last part $t$ is smaller than the previous smallest part because $s > t$ . The first $t$ parts of $β (λ)$ are consecutive and its smallest part is exactly $t$ , so $β (λ)$ falls under the first construction with parameter $s = t$ , and $α (β (λ)) = λ$ . Thus steps 2.1 and 2.2 define a sign-reversing involution on all nonexceptional nonempty distinct partitions.

2.3step 1.1algebra

The empty partition contributes the constant term $1$ . The only nonempty distinct partitions excluded from step 1.1 are the two staircase families $(2 k - 1, 2 k - 2, \dots, k)$ and $(2 k, 2 k - 1, \dots, k + 1)$ . Their sizes are $k + (k + 1) + \dots + (2 k - 1) = k (3 k - 1) / 2$ and $(k + 1) + (k + 2) + \dots + 2 k = k (3 k + 1) / 2$ , and each has exactly $k$ parts, so each contributes the sign $(- 1)^{k}$ .

3.1F2step 2.2step 2.3∎

By step 2.2, all nonexceptional nonempty distinct partitions cancel in opposite-parity pairs. Step 2.3 leaves only the empty partition and the two exceptional staircase families, so [F2] gives $\prod_{m \geq 1} (1 - x^{m}) = 1 + \sum_{k \geq 1} (- 1)^{k} (x^{k (3 k - 1) / 2} + x^{k (3 k + 1) / 2})$ , equivalently the two-sided sum over $r \in Z$ .

Depends on

Used by

Euler's pentagonal recurrence for partition numbers Corollary

Dependency tree · two levels

6 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 S19 Algebraic Combinatorics, Lecture 21: Partition theory (cont.). Franklin's combinatorial proof of Euler's pentagonal number theorem and more (standard reference, not scraped)
Darij Grinberg, Enumerative Combinatorics: class notes (standard reference, not scraped)