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Rains’ conjecture for Koornwinder moments

For every partition λ\lambda, specialize the Koornwinder moment indexed by λ\lambda to t=qt=q, normalize it as in Rains' formulation, and write the resulting rational function in reduced form. The conjecture asserts that its minimal numerator Nλ(q;a,b,c,d)N_\lambda(q;a,b,c,d) belongs to Z≥0[q,a,b,c,d]\mathbb{Z}_{\ge 0}[q,a,b,c,d]; equivalently, all coefficients of the minimal numerator are nonnegative integers.

References

Primary source

arXiv

Additional references

combinatoricsspecial-functionssymmetric-functions
Posted · edited

Progress summary

Refreshed
Claimed progress

A 2026 preprint proves the previously open one-column case, but the full conjecture for arbitrary partitions remains open.

Rains’ conjecture predicts coefficientwise positivity for normalized Koornwinder moments indexed by partitions. The full partition-valued statement remains broader than the newly established one-column specialization.

Known results

  • The one-row case was previously verified using rhombic staircase tableaux and its connection with the two-species ASEP (2026 source summarizing earlier work).
  • A 2015 treatment formulated the general conjecture and established the one-row/ASEP positivity framework, while leaving the general case open.

2026 one-column result

Younggwang Cho, Donghyun Kim, and Jang Soo Kim report a proof of the previously open one-column case for rescaled Askey–Wilson polynomial coefficients. A related 2026 preprint derives formulas for MλZM^Z_\lambda and proves restricted specializations (ξ,q)=(1,0)(\xi,q)=(1,0) and (ξ,q)=(1,1)(\xi,q)=(1,1), but not the full conjecture.

Current status (as of September 2026): the one-row and one-column cases are claimed established, while the full conjecture for arbitrary partition λ\lambda remains open.

Sources

Solutions 0

No solutions have been posted yet.