We present practical algorithms for ranking k-ary necklaces and Lyndon words of length n. The algorithms are based on simple counting techniques. By repeatedly applying the ranking algorithms, both necklaces and Lyndon words can be efficiently unranked. Then, explicit details are given to rank and unrank the length n substrings of the lexicographically smallest de Bruijn sequence of order n.

We consider the following two classes of simple graphs: open necklaces and closed necklaces, consisting of a finite number of cliques of fixed orders arranged in path-like pattern and cycle-like pattern, respectively. In these two classes we determine those graphs whose index (the largest eigenvalue of the adjacency matrix) is maximal.

We introduce and characterize extremal 2-cuts for good fractal necklaces. Using this characterization the related topological properties of 2-cuts, we prove that every necklace has a unique IFS in certain sense. Also, two necklaces admit only rigid homeomorphisms thus group self-homeomorphisms is countable. In addition, weaker co-Hopfian property also obtained.

This paper is devoted to the random generation of particular colored necklaces for which the number of beads of a given color is constrained (these necklaces are called v-balanced). We propose an efficient sampler (its expected time complexity is linear) which satisfies the Boltzmann model principle introduced by Duchon, Flajolet, Louchard and Schaeffer (7). Our main motivation is to show that ...

This paper describes a newly discovered aspect of T-codes, a family of variable length codes first described by Titchener in the 1980s. We show that the constant-length subsets of codewords generated during the Tcode construction process constitute sets of necklaces. Moreover, we show that these sets are complete except for a small subset which we characterize. Bounds on the number of constant-...

It is known that there are no more Lyndon words of length n than there are periodic necklaces of same length. This paper considers a similar problem where, additionally, the necklaces must be without some forbidden factors. This problem relates to a different context, concerned with the behaviours of particular discrete dynamical systems, namely, Boolean automata networks. A formal argument sup...