On a family of universal cycles for multi-dimensional permutations

Kitaev, Sergey and Qiu, Dun (2024) On a family of universal cycles for multi-dimensional permutations. Discrete Applied Mathematics, 359. pp. 310-320. ISSN 0166-218X

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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.

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