Ding–Ding antichain-polynomial Conjecture 4.2

For every $k,n\ge 1$ , let $A_{k,n}^{(1)}(x)$ and $A_{k,n}^{(2)}(x)$ denote each specified pair of antichain-polynomial sums associated with $[k]\times Q$ , where $Q=[2]\times[n]$ . The conjecture asserts that every such pair has a common interlacer: there exists a real-rooted polynomial $h_{k,n}(x)$ that interlaces both $A_{k,n}^{(1)}(x)$ and $A_{k,n}^{(2)}(x)$ . The supplied sources do not give the explicit definitions of the two antichain-polynomial sums.

References

Primary source

arXiv

Additional references

A common interleaver for two antichain polynomials on [k] × P_{n,s} — arXiv — Jian Ding, Lingen Ding

algebraic-combinatorics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture for two-row shapes, but the result has not yet been peer reviewed.

Ding and Dong posed Conjecture 4.2 in 2019 for $Q=[2]\times[n]$ : specified pairs of antichain-polynomial sums should have common interleavers. They noted that this would imply real-rootedness of the corresponding antichain polynomial.

Known results

The 2019 paper states that Conjecture 4.2 would imply real-rootedness of $\mathcal{N}_{[2]\times[n]\times[k]}(x)$ , formulated there as Conjecture 4.3 (Ding and Dong, 2019).

September 2026 claimed proof

A September 22, 2026 report says Jian Ding and Lingen Ding establish real stability for adjacent-shape bivariate polynomials and apply the Chudnovsky–Seymour compatibility criterion to obtain the required common interleaver, thereby proving Conjecture 4.2 for two-row Ferrers shapes. The associated preprint is not yet peer reviewed.

Current status (as of September 2026): Conjecture 4.2 is claimed proved for the stated two-row setting, but independent verification remains outstanding.

Sources

No solutions have been posted yet.