Catalan Formulas for Ekedahl-Oort Intersection Cohomology

Definitions and mathematical statement

The supplied record refers to Ekedahl--Oort spaces $\operatorname{EO}(r,h)$, but does not define these spaces, the admissible parameter pairs, or the local intersection-cohomology invariant to be computed. Let $\mathcal P$ denote the source's intended set of admissible pairs $(r,h)$, and let $\mathcal L_{r,h}$ denote its intended local intersection-cohomology data for $\operatorname{EO}(r,h)$. These are placeholders for missing source definitions, not definitions of the geometric objects or their invariants.

Source-intended conjectural task. Generalize the method referred to in the source so as to construct, for every $(r,h)\in\mathcal P$, an explicit expression for $\mathcal L_{r,h}$ in terms of the Catalan combinatorics intended there.

The supplied original wording conjectures explicit formulas for all $\operatorname{EO}(r,h)$ using "certain Catalan combinatorics"; the supplied extracted statement identifies local intersection cohomology as the target. Neither the method nor the Catalan objects, statistics, or formula conventions are specified. The base field, characteristic, coefficient system, normalization, and points or strata at which local intersection cohomology is sought are also unspecified. Consequently, this is an explicitly incomplete formulation of the intended task; a fully precise mathematical statement requires the missing source context.

Short English statement

The intended problem is to extend an existing method to express the local intersection cohomology of every admissible Ekedahl--Oort space $\operatorname{EO}(r,h)$ through explicit Catalan combinatorics. The supplied record does not specify the spaces, parameter restrictions, method, or combinatorial formula, so these require clarification from the source before mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006272$, exact source problem code catalogue entry, title: Catalan Formulas for Ekedahl--Oort Intersection Cohomology. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2025; no individual proposer is supplied.
• Enveloping Algebras and Geometric Representation Theory (2025), Oberwolfach Reports, pp. 1271--1332, DOI 10.4171/owr/2025/25. Supplied report provenance: https://doi.org/10.4171/owr/2025/25. No page specific to the conjecture or speaker attribution is supplied.
• Supplied literature source, labelled "OWR 2025 report": https://ems.press/content/serial-article-files/51851.
• Supplied literature source, labelled "Fang--Reineke, local intersection cohomology": https://arxiv.org/abs/2502.07688. The supplied record does not give a full bibliographic citation or identify the precise results used.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially solved and states that local intersection-cohomology and irreducible-component formulas are established, while the proposed Catalan-combinatorics formula for all $\operatorname{EO}(r,h)$ cases remains conjectural. This is supplied dated curation, not independently verified here; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and notation repaired. These are supplied verification claims, not independent verification performed for this draft.