12 problems

A latin hypercube of order $n$ and dimension $d$ is a $d$-dimensional array filled with $n$ symbols so that every line contains all symbols exactly once. A transversal is a diagona…

Let $M(d,n)$ denote the set of Latin hypercubes of dimension $d$ and order $n$. For all even $d$ and even $n>2$, let $H\in M(d,n)$ be a hypercube constructed by replacing the order…

Let $M(d,n)$ denote the set of Latin hypercubes of dimension $d$ and order $n$. A transversal is a set of $n$ entries containing each value in every coordinate position and each sy…

Reducibility conjecture. Every $d$-dimensional latin hypercube of order $n$ with the maximum number of cuboctahedra is reducible.

Let $\mathrm{AT}_3(k)$ denote the signed sum of $k\times k\times k$ Latin cubes, equivalently the evaluation of the fundamental invariant at the unit tensor. Three-dimensional Alon…

A $d$-dimensional permutation of order $n$ is a $d$-dimensional $0$-$1$ array in which every line contains exactly one $1$. Its permanent counts the transversals of the correspondi…

A latin $n$-cuboid of order $q$ is an $n$-dimensional array with one dimension of length less than $q$ and the remaining dimensions of length $q$, whose lines contain different sym…

For $d\in\mathbb N$, a $d$-dimensional order-$n$ Latin hypercube is a $d$-dimensional array indexed by $[n]^d$ with entries from $[n]$, such that every line contains each symbol ex…

Let $\ell$ be the number of coefficients in the product-null hypothesis … and suppose the corresponding asymptotically normal standardized estimator is available. For unit-fraction…

A latin hypercube of dimension $d$ and order $n$ is the $d$-dimensional analogue of a latin square, with every symbol occurring exactly once on each axis-parallel line. Sun's conje…

Let $Q^n(\mathbb{Z}_4)$ and $Q^n(\mathbb{Z}_2^2)$ denote the $n$-dimensional latin hypercubes isotopic to the Cayley tables of the corresponding iterated groups. A transversal is a…

Asymptotic enumeration conjecture.