## Abstract

A universal cycle (u-cycle) for permutations of length $n$ is a cyclic word, any size $n$ window of which is order-isomorphic to exactly one permutation of length $n$, and all permutations of length $n$ are covered. It is known that u-cycles for permutations exist, and they have been considered in the literature in several papers from different points of view.

In this paper, we show how to construct a family of u-cycles for multi-dimensional permutations, which is based on applying an appropriate greedy algorithm. Our construction is a generalisation of the greedy way by Gao et al.¥ to construct u-cycles for permutations. We also note the existence of u-cycles for $d$-dimensional matrices.

In this paper, we show how to construct a family of u-cycles for multi-dimensional permutations, which is based on applying an appropriate greedy algorithm. Our construction is a generalisation of the greedy way by Gao et al.¥ to construct u-cycles for permutations. We also note the existence of u-cycles for $d$-dimensional matrices.

Original language | English |
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Pages (from-to) | 310-320 |

Journal | Discrete Applied Mathematics |

Volume | 359 |

Early online date | 24 Aug 2024 |

Publication status | E-pub ahead of print - 24 Aug 2024 |

## Keywords

- universal cycle
- combinatorial generation
- greedy algorithm
- multi-dimensional permutation
- multi-dimensional matrix