Combinatorics of Signed Adjacency Polytopes

Definitions and mathematical statement

Use the conventional dimension range $n\geq 1$, with $n$ an integer; the supplied record does not state this range explicitly. For every sign sequence $\varepsilon=(\varepsilon_1,\ldots,\varepsilon_{n-1})\in\{-,+\}^{n-1}$, define \[ P_{\varepsilon}=\left\{x\in{\mathbb{R}}^n:\begin{array}{ll} 0\leq x_j\leq 1 & (1\leq j\leq n),\\ x_i+x_{i+1}\leq 1 & (1\leq i\leq n-1,\ \varepsilon_i=-),\\ x_i+x_{i+1}\geq 1 & (1\leq i\leq n-1,\ \varepsilon_i=+) \end{array}\right\}. \] All inequalities include their boundary points. The adjacency constraints are indexed by $1\leq i\leq n-1$, with no constraint joining $x_n$ to $x_1$. For $n=1$, the sign sequence is empty and the adjacency constraints are vacuous.

With respect to the ambient lattice ${\mathbb{Z}}^n$, use the standard Ehrhart convention: for each integer $m\geq0$, let $L_{P_{\varepsilon}}(m)=\#(mP_{\varepsilon}\cap{\mathbb{Z}}^n)$, where $mP_{\varepsilon}=\{mx:x\in P_{\varepsilon}\}$. The $h^*$-polynomial is characterized by \[ \sum_{m=0}^{\infty}L_{P_{\varepsilon}}(m)t^m =\frac{h^*_{P_{\varepsilon}}(t)}{(1-t)^{n+1}}, \qquad h^*_{P_{\varepsilon}}(t)=\sum_{k=0}^{n}h^*_{\varepsilon,k}t^k. \] Problem. For every such $n$ and $\varepsilon$, give a combinatorial interpretation of $h^*_{P_{\varepsilon}}(t)$, explaining its coefficients in terms of combinatorial objects or statistics associated with the sign sequence. The supplied original wording additionally asks to determine the polynomials' "structure," but specifies no particular structural property; that additional request remains imprecise.

The supplied registry labels this problem open and labels the statement corrected_verified. These are supplied curation claims, not an independent verification. The accompanying dated literature assessment instead describes a record containing only the words "open problem," which conflicts with the substantive statement reproduced here.

Short English statement

For each sequence of signs, impose the corresponding adjacent-coordinate sum inequalities on the unit cube. Find combinatorial objects or statistics that explain the coefficients of the resulting polytope's Ehrhart $h^*$-polynomial. The additional request for its "structure" is unspecified, and the supplied literature assessment conflicts with the supplied mathematical statement.

Sources

• Local UnsolvedMath registry, permanent ID $30003813$, title "Combinatorics of Signed Adjacency Polytopes," source problem code catalogue entry, in the supplied Oberwolfach Reports Open Problems collection. No proposer is identified; the supplied proposed year is 2018.
• Supplied report citation: Enumerative Combinatorics (2019), Oberwolfach Reports, pp. 1381--1464, DOI 10.4171/owr/2018/23. Supplied source URL: https://doi.org/10.4171/owr/2018/23. The relationship between the citation's 2019 label and its 2018 DOI and proposed-year metadata is unresolved.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2018/23.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. It labels the record open for triage because it purportedly contains only "open problem" and no testable mathematical proposition. This characterization conflicts with the supplied original and cleaned statements. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field claims that the official report was checked directly and that the cleaned statement restores definitions while preserving the target. That verification claim is attributed to the supplied registry; no independent source consultation is asserted here.