Athanasiadis–Douvropoulos–Kalampogia-Evangelinou conjecture on noncrossing-partition h-polynomials

For every finite Coxeter group $W$ of rank $r$ , the $h$ -polynomial of the order complex of its noncrossing-partition lattice admits a nonnegative real-rooted symmetric decomposition with respect to $r-1$ : there exist polynomials $a_W(t),b_W(t)$ with nonnegative coefficients such that $h(\Delta(\mathrm{NC}_W),t)=a_W(t)+t b_W(t)$ , $a_W(t)=t^{r-1}a_W(t^{-1})$ , $b_W(t)=t^{r-2}b_W(t^{-1})$ , and both $a_W(t)$ and $b_W(t)$ are real-rooted.

Progress summary

Solved

A 2026 preprint claims to settle the conjecture, but the claim has not yet been independently verified.

Athanasiadis, Douvropoulos, and Kalampogia-Evangelinou studied real-rootedness and symmetric decompositions of $h$ -polynomials for noncrossing-partition lattices of finite Coxeter groups. Their 2023 preprint established major cases and results for the associated lattices; a broader 2026 preprint now claims the conjecture is resolved.

Known results

Every finite Coxeter group $W$ has a noncrossing-partition lattice with real-rooted chain polynomial (Athanasiadis, Douvropoulos, and Kalampogia-Evangelinou, 2023).
The order-complex polynomial $h(\Delta(NC^W),x)$ is real-rooted for every finite Coxeter group $W$ (2023).
For irreducible $W$ , this polynomial has a nonnegative real-rooted symmetric decomposition with respect to $r_W-1$ , implying unimodality (2023).

August 2026 claimed resolution

Gao, Liu, Yang, and Zhao claim a parking-function interpretation for a special Lascoux series and interlacing symmetric decompositions for $k$ -divisible noncrossing-partition posets, implying the conjecture across finite Coxeter types. The result appears only as an unrefereed preprint, so its proof has not been independently confirmed.

Current status (as of August 2026): The 2023 real-rootedness results are established in the cited preprint, while the broader conjecture is claimed solved by the 2026 preprint but remains unverified.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Lascoux series, parking functions and noncrossing partitions — arXiv — Alice L. L. Gao, Xin-Bei Liu, Arthur L. B. Yang, James J. Y. Zhao

algebraic-combinatorics coxeter-groups

2 days ago