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bibkey: gottlieb2025pnim authors: Eric Gottlieb; Matjaž Krnc; Peter Muršič year: 2025 title: Nim on Integer Partitions and Hyperrectangles doi: 10.48550/arXiv.2506.04991 url: https://arxiv.org/abs/2506.04991 claim: Section 1.2 defines PNim row and column subset deletion; section 2 defines Young’s order; section 3 and Proposition 2 define heaviness; printed Conjecture 2 predicts a heavy interval below any heavy rectangle. strata_touched:

  • D5/S0/Certificates/Games/PnimHeavyIntervalRefutation license: citation-only triage: anchor

PNim and the heavy-interval conjecture

In single-partition PNim, a move deletes a nonempty subset of diagram rows or columns, merging the remaining rows or columns. Partitions have positive nonincreasing parts. Young’s order compares lengths and aligned parts. The Sprague-Grundy value is the least excluded follower value.

Proposition 2 gives the longest-play length of a nonempty partition as its first part plus its number of parts minus one. A partition is heavy when its Sprague-Grundy value equals this length. Proposition 1 states the corresponding upper bound, not the longest-play identity.

Printed Conjecture 2, page 14, asserts that if [(a+1)^(b+1)] is heavy, then every partition between [a+1,a,…,a-b+1] and that rectangle in Young’s order is heavy. Both endpoints are partitions exactly when the integer parameters satisfy a >= b >= 0. This note attests the printed definitions and assertion, not the assertion’s truth. The formal refutation uses a=8, b=7 and [9,9,8,8,8,5,5,5], whose Grundy value is 3 rather than its longest-play length 16; the upper rectangle is heavy. No priority or corrected conjecture is asserted.

The arXiv API queries on 2026-09-17 returned one PNim paper (this v1), one matching title, and six papers matching Gottlieb and Krnc. These bounded searches do not establish absence of a settlement in journal versions, non-arXiv manuscripts, or later sources.

Verified locator

  • DOI: https://doi.org/10.48550/arXiv.2506.04991
  • URL: https://arxiv.org/abs/2506.04991
  • Version: https://arxiv.org/pdf/2506.04991v1 (2025-06-05).
  • Scope: section 1.2 (PNim moves and Theorem 1), section 2 (Young’s order), section 3 (Proposition 2 and heaviness), and printed page 14 (Conjecture 2).