bibkey: wiseman2025disjoint authors: Gus Wiseman year: 2025 title: OEIS A384350 and A384318 — disjoint strict partition families doi: null url: https://oeis.org/A384350 claim: The entries conjecture that an outside subset sum characterizes nonuniqueness of disjoint strict partition families. strata_touched:

• D5/S3/ArithSums/DisjointStrictRefinement license: citation-only triage: anchor

Disjoint strict partition families

Verified locator

doi: null url: https://oeis.org/A384350

The COMMENTS in https://oeis.org/A384350/internal and https://oeis.org/A384318/internal label the characterization by more than one disjoint strict partition family a conjecture. A384350 counts subsets of an initial interval; A384318 counts strict partitions of a prescribed total.

The NAME in https://oeis.org/A384322/internal and the FORMULA in https://oeis.org/A384317/internal state the correspondence without a proof. These sources establish the question and its terminology.

Mathematical scope

For a finite positive set S, a family chooses one positive strict integer partition for each member, with that member as the sum of its block. All blocks are pairwise disjoint. The singleton family is always available; nonuniqueness means that at least one block can be changed.

The repository proof uses the least changed member. Every part in its block is smaller than that member. Any such part in S would retain its own singleton block, contradicting disjointness. Conversely, an outside subset sum permits replacement of one singleton block. This is a pointwise statement for every S; no sequence coefficients or independent product of block counts are asserted.

Proof scope

Proposition 1 on p.3 of arXiv:2601.10227 and p.2 of arXiv:2206.04261 discuss reducing an already refinable part to two missing parts; they do not prove the disjoint-family implication used here.

No exhaustive literature absence or global novelty is claimed. Unopened paper content remains ASSUMED-UNVERIFIED.