Gray–Payne–Watson and Gray–Payne–Swisher–Watson partition conjectures

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ range over partitions into distinct parts, and define its perimeter by $\operatorname{per}(\lambda)=\lambda_1+\ell$ . Let $r_o(n)$ be the number of such partitions with $\operatorname{per}(\lambda)=n$ and with more odd parts than even parts, and let $r_e(n)$ be the number with $\operatorname{per}(\lambda)=n$ and with more even parts than odd parts. The conjecture is that $r_o(n)>r_e(n)$ for every integer $n\geq 9$ .

References

Primary source

arXiv

Additional references

A perimeter analogue of Franklin's identity and an inequality related to the parity of parts — arXiv — Philip Cuthbertson

analytic-combinatorics partition-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

Two new unrefereed manuscripts claim to settle both partition conjectures, but their proofs have not been independently verified.

The conjectures concern fixed-perimeter partition statistics: a Franklin-type comparison between two multiplicity classes and an odd-versus-even bias for distinct-part partitions. Gray, Payne, Swisher, and Watson formulated the latter in 2026; the manuscripts claim both conjectures are now proved.

Known results

Banerjee, Bhattacharjee, Dastidar, and Saikia (2022) proved the earlier fixed-size parity inequality conjectured by Kim, Kim, and Lovejoy.
Gray, Payne, Swisher, and Watson (2024) established the general fixed-perimeter bias $r_e(n)<r_o(n)$ for $n\ne 2$ and a fixed-perimeter Euler-type identity.

September 2026 claimed proofs

On September 10, 2026, Marcus McCrea’s manuscript claimed an asymptotic proof of the Franklin-statistics conjecture for $k\geq 3$ , with $k=2$ recovering a known identity. A separate 2026 manuscript states that the distinct-part bias conjecture $rd_o(n)>rd_e(n)$ for $n\geq 9$ is true and supplies explicit mappings.

Current status (as of September 2026): both conjectures have claimed manuscript proofs, but neither claim has independent verification; absent that, their mathematical status remains unconfirmed.

Sources

No solutions have been posted yet.