Universal cycles for weak orders (Q2870511)

Given a set \({\mathcal C}\) of strings, all of the same length, a universal cycle (ucycle) is a cyclic word that contains each element of the set \({\mathcal C}\) exactly once. Examples of ucycles include de Bruijn cycles and Gray codes, first introduced by \textit{F. Chung} et al. [Discrete Math. 110, No. 1--3, 43--59 (1992; Zbl 0776.05001)]. A further generalization of the notion of a ucycle is the notion of an \(s\)-overlap cycle, first introduced in [\textit{A. P. Godbole} et al., Congr. Numerantium 204, 161--171 (2010; Zbl 1229.05104)]. This paper studies weak orders, defined as transitive and complete relations. The authors prove the existence of universal and \(s\)-overlap cycles for weak orders, as well as for weight orders of fixed height or weight.