bibkey: kravitz2024rearranging
authors: Noah Kravitz
year: 2024
title: Rearranging small sets for distinct partial sums
doi: 10.48550/arXiv.2407.01835
claim: Theorem 1.3 and Remark (1) construct two-sided valid orderings of every finite subset of the nonzero integers.
strata_touched:
  - D5/S1/Words/Sequenceability/IntegerStrongOrdering
license: citation-only
triage: anchor

Rearranging small sets for distinct partial sums

Theorem 1.3 of arXiv:2407.01835v2 proves the integer ordering theorem. Its proof on page 2 simultaneously orders two finite sets of positive integers so their prefix sums intersect only at the empty pair and, possibly, the pair of complete lists. Induction removes an element whose deletion avoids a collision of the two totals. Reversing the positive ordering and negating the other ordering gives the integer word.

Remark (1), page 3, states that these orderings are two-sided, equivalently that they have no zero-sum proper consecutive subordering. The complete word may sum to zero. Kravitz attributes a similar two-step route to Will Sawin’s 2015 MathOverflow contribution, reference [16]; this is Kravitz’s attribution, not an independent verification of that post.

This note attests the integer result and its proof, without a novelty claim. It does not attest a solution for arbitrary abelian groups, an integral primitive certificate, or the ordered-kernel extension of Remark (2).

Verified locator

• Version: https://arxiv.org/abs/2407.01835v2
• DOI: https://doi.org/10.48550/arXiv.2407.01835
• Sawin reference as cited by Kravitz: https://mathoverflow.net/q/202857