bibkey: ballantine2024elementary authors: Cristina Ballantine, George Beck, Mircea Merca, Bruce E. Sagan year: 2024 title: Elementary symmetric partitions doi: 10.48550/arXiv.2409.11268 url: https://arxiv.org/abs/2409.11268v3 claim: Conjecture 16 gives a residue-class table for r_{d,2}(n) minus r_{d,3}(n) when d is 2, 3, 4, or 5. strata_touched:

• D5/S1/Recurrence/Partitions/BallantineRegularSymmetricImageRefutation license: citation-only triage: anchor

Elementary symmetric partitions

Ballantine, Beck, Merca and Sagan define the elementary symmetric image of a partition and count its regular members. Section 5, Conjecture 16 on printed page 19 gives four residue-class columns for the difference between the degree-two and degree-three counts.

The literal table is false at d = 5 and n = 4. The degree-two image set is {(3), (4), (2,2,1), (1,1,1,1,1,1)}, whose four members are all 5-regular. The degree-three image set is {(2), (1,1,1,1)}, whose two members are also 5-regular. Their count difference is two, while the residue-four cell of the printed table gives one.

Version 3, dated October 25, 2024, retains Conjecture 16 and has no journal reference on arXiv. Searches recorded with the source found no later paper settling Conjecture 16. Google Scholar and Semantic Scholar coverage was not verified.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2409.11268
• URL: https://arxiv.org/abs/2409.11268v3
• Version and location: arXiv:2409.11268v3, printed pages 2–3 and 19.