TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)audited 2026-08-28

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Partitions with k parts are equinumerous with partitions whose largest part is k

Statement

For every $n \geq 0$ and $k \geq 0$ , conjugation is a bijection between:

partitions of $n$ with exactly $k$ parts, and
partitions of $n$ whose largest part is $k$ .

In particular, these two sets have the same cardinality.

Facts & Assumptions

Given: integers $n \geq 0$ and $k \geq 0$ .

[F1]

In a Ferrers diagram, the number of rows is the number of parts and the number of cells in the top row is the largest part (Ferrers and Young diagrams, conjugate partitions, self-conjugacy, and the Durfee square, The functions p(n), p_k(n), and the standard restricted partition families).

[L1]

Partition conjugation is an involution (Conjugating a partition twice returns the original partition).

Proof

technique · bijection

1.1

If $k = 0$ and $n > 0$ , then there is no partition of $n$ with exactly $0$ parts and no partition of $n$ whose largest part is $0$ , so both displayed sets are empty. If $k = 0$ and $n = 0$ , then both displayed sets consist only of the empty partition; by the partition convention recalled in [F1], its largest part is $0$ . Thus the claim holds when $k = 0$ . Assume now $k \geq 1$ . Let $λ$ be a partition of $n$ with exactly $k$ parts. Then its Ferrers diagram has exactly $k$ rows by [F1]. After transposition, the conjugate diagram has top row length $k$ , because the first column of the original diagram had one cell in each of the $k$ rows. Thus $λ^{'}$ has largest part $k$ . The same argument in reverse shows that any partition with largest part $k$ conjugates to one with exactly $k$ parts.

2.1

Step 1.1 shows that conjugation maps each of the two displayed sets into the other, and [L1] shows that this map has its own inverse. Therefore it is a bijection between them, so the two sets have equal cardinality.

step 1.1L1∎

Depends on

Used by

Partitions with at most k parts are equinumerous with partitions whose parts are all at most k Corollary

Dependency tree · two levels

5 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

Stephen Melczer, An Invitation to Enumeration, Chapter 9: Integer Partitions (standard reference, not scraped)
Darij Grinberg, Enumerative Combinatorics: class notes (standard reference, not scraped)