Enumeration and Structure of Higher-Dimensional Permutations

Definitions and mathematical statement

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. For an integer $d\geq 3$, a $d$-dimensional permutation of order $n$ is a function \[ A:[n]^{d+1}\longrightarrow\{0,1\} \] such that, for every $j\in\{1,\ldots,d+1\}$ and every fixed choice of $i_k\in[n]$ for $k\ne j$, \[ \sum_{t=1}^{n}A(i_1,\ldots,i_{j-1},t,i_{j+1},\ldots,i_{d+1})=1. \] Thus every axis-parallel line contains exactly one entry equal to $1$. Let $P_d(n)$ denote the number of these functions, counted as arrays on the specified indexed domain $[n]^{d+1}$.

The supplied clean statement asks whether, for every fixed integer $d\geq3$, \[ P_d(n)=\left((1+o(1))\frac{n}{e^d}\right)^{n^d} \qquad(n\to\infty), \] where $n$ tends to infinity through positive integers, $e$ is the base of the natural logarithm, and the $o(1)$ term may depend on the fixed dimension $d$. Equivalently, is there, for each such $d$, a real sequence $\varepsilon_d(n)\to0$ such that \[ P_d(n)=\exp\!\left(n^d\left(\log n-d+\varepsilon_d(n)\right)\right)? \]

The original source-record wording asks more broadly for an appropriate higher-dimensional analogue of a permutation and its basic enumeration and structural properties. The displayed question formalizes the specific enumeration target in the supplied clean statement; the broader structural request supplies no precise additional assertion to formalize. The record labels the problem open and the clean statement corrected_verified, and reports verification against the official report. These are supplied curation claims, not independent verification by this draft.

Short English statement

Count the zero--one arrays on $[n]^{d+1}$ having exactly one $1$ on every axis-parallel line. For each fixed integer $d\geq3$, does their number equal $(n/e^d)^{n^d}\exp(o(n^d))$ as $n\to\infty$? The original broader request about structural properties remains unspecified.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30003332$, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems; title: Enumeration and Structure of Higher-Dimensional Permutations. No named proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Combinatorics (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/1, pp. 5--81. Source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2017/1. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2018 label and its 2017 DOI component are retained without resolving their relationship.
• Supplied literature-source link labelled "Linial--Luria, high-dimensional permutations": https://arxiv.org/abs/1106.0649.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes high-dimensional permutations as natural higher-dimensional Latin objects, and reports that a matching lower bound for the central asymptotic enumeration problem is lacking in dimensions at least three. This is supplied dated literature triage derived from the OWR curation, not independently verified here; its time-sensitive status and cited claims require independent checking.