Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)audited 2026-08-28
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The unlabelled-to-unlabelled cells of the twelvefold way

Statement

Fix positive integers n and k. For placements of n indistinguishable balls into k indistinguishable boxes:

  1. the arbitrary placements are counted by the partitions of n with at most k parts, equivalently by the partitions of n whose parts are all at most k;
  2. the injective placements are counted by 1 when n≤k and by 0 when n>k;
  3. the surjective placements are counted by pk(n).

Facts & Assumptions

Given: positive integers n and k.

[L1]

Under the page conventions, an unlabelled-to-unlabelled placement is encoded by its nonzero occupancies written in nonincreasing order (Conventions for integer partitions, Ferrers diagrams, and the twelvefold-way table).

[L2]

Partitions with at most k parts are equinumerous with partitions whose parts are all at most k (Partitions with at most k parts are equinumerous with partitions whose parts are all at most k).

Proof

technique · classification
1.1

By [L1], an arbitrary placement is determined by the list of its positive occupancies, written in nonincreasing order. These occupancies sum to n, so they form a partition of n; because there are only k boxes, there can be at most k positive occupancies. Conversely, any partition of n with at most k parts becomes such a placement by reading its parts as the nonzero box occupancies. This proves clause 1 in its "at most k parts" form.

L1construct
2.1

A placement is surjective exactly when every box is occupied, so there are exactly k positive occupancies. By step 1.1 these are exactly the partitions of n into k positive parts, which are counted by pk(n). This is clause 3.

step 1.1
2.2

A placement is injective exactly when every occupied box contains one ball. Therefore the positive occupancy list must be (1,…,1) with n entries. Such a list exists exactly when n≤k, and when it exists it is unique. This proves clause 2.

step 1.1
3.1

The second description in clause 1 follows from [L2].

step 1.1L2∎

Depends on

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Sources