Nonnegative Coefficients for Cylindric Order Polynomials

Definitions and mathematical statement

Write $\sigma=\lambda/\mu/d$ for a cylindric shape in the sense intended by the supplied source, and write $F_\sigma(n)=\operatorname{CPP}_{\lambda/\mu/d}(n)$ for its associated expression. The supplied record does not define cylindric shapes, the admissibility conditions on $\lambda,\mu,d$, the function $\operatorname{CPP}$, or its domain in $n$. These conventions therefore remain unresolved; no particular counting or order-polynomial convention is imposed here. Let $D_\sigma$ denote that unspecified intended domain.

Conjecture 4, as reported. For every admissible cylindric shape $\sigma=\lambda/\mu/d$, there exist an integer $m\geq 0$ and real numbers $a_{\sigma,0},\ldots,a_{\sigma,m}\geq 0$ such that \[ P_\sigma(t)=\sum_{k=0}^{m}a_{\sigma,k}t^k, \qquad \operatorname{CPP}_{\lambda/\mu/d}(n)=P_\sigma(n) \quad\text{for every }n\in D_\sigma. \] Thus the asserted coefficients are nonnegative in the ordinary monomial basis. This is a conditional transcription of the intended assertion: its mathematical scope cannot be fixed completely until the missing definitions and domain are supplied.

The excerpt attributes the conjecture to its reference $[5]$ and reports that the circular fence poset case was settled in $[5,\mathrm{Proposition}\ 7.6]$. It also states that, by the general strategy in the referenced paper, it would suffice to establish nonnegativity of the linear coefficient of $\operatorname{CPP}_{\lambda/\mu/d}(n)$. The relevant strategy, hypotheses, and identification of $[5]$ are absent, so this reported reduction is not asserted here as a standalone theorem.

Short English statement

The conjecture asks whether the expression associated with every cylindric shape is a polynomial in $n$ whose ordinary monomial coefficients are all nonnegative. The source does not supply the definitions of the shapes or $\operatorname{CPP}$, or the allowed values of $n$; these must be recovered before a fully specified statement is possible. The record labels the problem open and reports a settled circular fence special case, subject to the supplied dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006532$, title Nonnegative Coefficients for Cylindric Order Polynomials, exact source problem code catalogue entry. The record supplies status open and statement status corrected_verified; its verification note says only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. These are supplied registry labels, not independent verification.
• Enumerative Combinatorics (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/2, pp. 81--156. Source report and literature-source label: Oberwolfach Report 2026/2. https://doi.org/10.4171/owr/2026/2. The supplied excerpt calls the assertion Conjecture 4 and attributes it to reference $[5]$; the bibliography identifying that reference is not included. No proposer is named in this record.
• Supplied literature source: Kantarci Oguz--Ravichandran, Discrete Mathematics. https://doi.org/10.1016/j.disc.2022.113218.
• Supplied literature source: Skew shapes, Ehrhart positivity and beyond. https://arxiv.org/abs/2503.16403.
• Literature assessment dated exactly 2026-08-22: the supplied curation describes the full cylindric nonnegative-coefficient conjecture as still open in the report and says the literature settles special circular-fence cases. This is supplied dated curation derived from the Oberwolfach Reports material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent checking.