On shortening u-cycles and u-words for permutations

Abstract

This paper initiates the study of shortening universal cycles (ucycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for n-permutations exist of lengths n! + (1 − k)(n − 1) for k = 0, 1, . . . , (n − 2)!. 123 132 231 "12" 213 312 321 "21" Figure 1: Clustering the graph of overlapping permutations of order 3Removing the requirement for a u-cycle to be a cyclic word, while keeping the other properties, we obtain a universal word, or u-word. Of course, existence of a u-cycle u 1 u 2 · · · u N for n-permutations triv-

On a Greedy Algorithm to Construct Universal Cycles for Permutations

“…It would be interesting to extend our approach to generate universal cycles for permutations in the context of shortened universal words/cycles for permutations that were introduced in [10].…”

“…To illustrate the idea of one of the two ways suggested in [10] to shorten universal cycles for permutations, consider the word 112, which is claimed to be a universal cycle for all permutations of length 3, thus shortening a "classical" universal cycle for these permutations, say, 145243. Indeed, we can treat equal elements as incomparable elements, while the relative order of these incomparable elements to the other elements must be respected.…”

“…The main goal of [10] is to study compression possibilities for (classical) universal cycles for permutations. In particular, [10] shows that such universal cycles exist of lengths n! − kn for k = 0, 1, .…”

“…, (n − 1)!. The approach in [10] to obtain these results is graph theoretical, and it actually does not provide any explicit constructions of the objects in the general case. Thus, (a modification of) our approach to generate universal cycles could potentially be useful in generating shortened universal words/cycles, and this is an interesting direction of further research.…”

On universal partial words for word-patterns and set partitions

“…as an interval of consecutive letters). In particular, universal cycles for sets of words are nothing else but the celebrated de Bruijn sequences that have found widespread use in real-world applications (see the references in [6]). Examples of objects considered in [3] are permutations (that required a slight modification of the notion of a u-cycle) and set partitions (that admit encoding in terms of words).…”