## Abstract

An r×s partial Latin rectangle (l_{ij}) is an r×s matrix containing elements of {1,2,…,n}∪{⋅} such that each row and each column contain at most one copy of any symbol in {1,2,…,n}. An entry is a triple (i,j,l_{ij}) with l_{ij}≠⋅. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of m-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) r=s=n, i.e., partial Latin squares, (b) r=2 and s=n, and (c) r=2 and s≠n.

Original language | English |
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Pages (from-to) | 1242-1260 |

Number of pages | 19 |

Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2017 |

Externally published | Yes |

## Keywords

- Autoparatopism
- Autotopism
- Latin square
- Partial Latin rectangle
- Partial Latin rectangle graph