Enumerations of Universal Cycles for $k$-Permutations

TL;DR

This paper introduces a new enumeration method for universal cycles of k-permutations, providing exact formulas for cases k=2 and 3, advancing combinatorial understanding of these cyclic arrangements.

Contribution

The paper proposes a novel enumeration technique for universal cycles of k-permutations and derives precise formulas for specific cases, enhancing combinatorial enumeration methods.

Findings

Exact enumeration formulas for k=2 and 3 universal cycles.

Introduction of a new enumeration method for k-permutations.

Improved understanding of the structure of universal cycles.

Abstract

Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when $k=2,3$.