Define a Latin hypercube of order *n* and
dimension *d* to be a set of *n ^{d}* tuples
(

Latin hypercubes of dimension 1, 2 and 3 are commonly called
**permutations**, **Latin squares**, and **Latin cubes**,
respectively. The hypercube can be represented
as a *d*-dimensional array *C* with
*C*[x_{1},*x _{2}*,...,

0123 1032 2301 3210 |
1032 0123 3210 2301 |
2301 3210 0123 1032 |
3210 2301 1032 0123 |

C[i,j,0] |
C[i,j,1] |
C[i,j,2] |
C[i,j,3] |

A Latin hypercube is **reduced** if, whenever *d-1* entries
of a tuple are 0, the other two entries are equal. In the case
of Latin squares, this is equivalent to the first row and column
being in natural order. The example above is reduced.

Given a Latin hypercube
*A*={(*x _{1}*,...,

Given a Latin hypercube
*A*={(*x _{1}*,...,

The operation of combining a conjugacy and an isotopism is called a
**paratopism**. Paratopism is an equivalence operation whose
equivalence classes are called **main classes**.
A paratopism which takes a Latin hypercube
*A* onto itself is an **autoparatopism** of *A*. The
autoparatopisms of *A* form a group under composition, called
the **autoparatopy group** Par(*A*). The autotopy group
is a subgroup of the autoparatopy group.

The relative counts of Latin hypercubes and their isotopy or main classes depends on the group sizes:

- The total number of Latin hypercubes of order
*n*and dimension*d*is*n*! (*n*-1)!^{d-1}times the number of reduced Latin hypercubes. - The number of reduced Latin hypercubes in the isotopy class of
a Latin hypercube
*A*is*n*^{d-1}*n*! / |Is(*A*)|. - The number of isotopy classes in the main class of a Latin
hypercube
*A*is

(*d*+1)! |Is(*A*)| / |Par(*A*)|. - The number of reduced Latin hypercubes in the main class of a Latin
hypercube
*A*is

(*d*+1)!*n*!*n*^{d-1}/ |Par(*A*)|. - The total number of Latin hypercubes in the main class of a Latin
hypercube
*A*is

(*d*+1)! (*n*!)^{d+1}/ |Par(*A*)|.

For Latin squares, see the Latin squares page.
Here we give one representative from each main class of Latin hypercubes
for some small orders and dimensions. The files have one line for each
main class, containing a hypercube, then |Is(*A*)| and
|Par(*A*)|/|Is(*A*)|. The hypercube is given as a list
of the values
*C*[x_{1},*x _{2}*,...,

This data was computed by Brendan McKay and Ian Wanless.

n |
d |
reduced hypercubes |
isotopy classes |
main classes |

4 | 3 | 64 | 12 | 5 |

4 | 4 | 7132 | 328 | 26 |

4 | 5 | 201538000 | 2133586 | 4785 |

5 | 3 | 40246 | 59 | 15 |

5 | 4 | 31503556 | 5466 | 86 |

5 | 5 | 50490811256 | 1501786 | 3102 (gzipped) |

6 | 3 | 95909896152 | 5678334 | 264248 (gzipped) |

B. D. McKay and I. M. Wanless, A census of small Latin hypercubes,
*SIAM Journal on Discrete Mathematics* 22 (2008) 719-736.
preprint in PDF

Page Master: Brendan McKay, brendan.mckay@anu.edu.au