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this paper ## Sawada 23 developed an algorithm to generate k-ary bracelets in constant ## amortized time. Proskurowski et al. 17 show that the orbits of the ' Z. Z. Z . composition of b and d can be generated in amortized Oktime, which is CAT if k is fixed. It remains an interesting challenge to develop efficient algorithms for the other compositions

A k-ary necklace is an equivalence class of k-ary strings under rotation. A necklace of fixed density is a necklace where the number of zeroes is fixed. We present a fast, simple, recursive algorithm for generating (i.e., listing) fixed density k-ary necklaces or aperiodic necklaces. The algorithm is optimal in the sense that it runs in time proportional to the number of necklaces produced. 1 Introduction There are many reasons to develop algorithms for producing lists of basic combinatorial objects. First, the algorithms are truely useful and find many applications in diverse areas such as hardware and software testing, non-parametric statistics, and combinatorial chemistry. Secondly, the development of these algorithms can lead to mathematical discoveries about the objects themselves, either experimentally, or through insights gained in the development of the algorithms. The primary performance goal in an algorithm for listing a combinatorial family is an algorithm whose running tim...

Introduction Let A # fa 1 #a 2 #####a k gbe a set with k elements, and QhAi the associative (non-commutative) algebra on A. Defining a Lie bracket (#x# y##xy # yx)onthisQ-module turns it into a Lie algebra, and we will denote by L#A# its Lie subalgebra generated by A (i.e. L#A# is the free Lie algebra on A and QhAi its enveloping algebra). A will now be called an alphabet, whose elements are the letters,andA # is the free monoid (the set of all words) over A. We know that<F9.1

Given a totally ordered alphabet A, a Lyndon word is a word that is strictly smaller, for the lexicographical order, than any of its conjugates (i.e., all words obtained by a circular permutation on the letters). Lyndon words were introduced by Lyndon [6] under the name of “standard lexicographic sequences ” in order to give a base for the free Lie algebra over A. The set of Lyndon words is denoted by L. For instance, with a binary alphabet A = {a, b}, the first Lyndon words until length five are L = {a, b,ab, aab, abb, aaab, aabb, abbb, aaaab, aaabb, aabab, aabbb, ababb, abbbb,...}. Note that a non-empty word is a Lyndon word if and only if it is strictly smaller than any of its proper suffixes. The standard (suffix) factorization of Lyndon words plays a central role in this framework (see [5], [7], [8]). For w ∈ L \ A a Lyndon word not reduced to a letter, the pair (u, v) of Lyndon words such that w = uv and v of maximal length is called the standard factorization. The words u and v are called the left factor and right factor of the standard factorization. Equivalently, the right factor v of the standard factorization of a Lyndon word w which is not reduced to a letter can be defined as the smallest proper suffix of w for the lexicographical order. For instance we have the following standard factorizations: aaabaab = aaab · aab aaababb = a · aababb aabaabb = aab · aabb. One can then associate to a Lyndon word w a binary tree T(w) called its Lyndon tree recursively built in the following way: – if w is a letter, then T(w) is a leaf labeled by w, – otherwise T(w) is an internal node having T(u) and T(v) as children where u · v is the standard factorization of w.

Many applications call for exhaustive lists of strings subject to various constraints, such as inequivalence under group actions. A k-ary necklace is an equivalence class of k-ary strings under rotation (the cyclic group). A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algorithms for generating (i.e., listing) all necklaces and binary unlabeled necklaces. Generalization is made to the case where no substring 0 t occurs, for fixed t. All these algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. As an application, we describe the implementation of a fast algorithm for listing all degree n irreducible and primitive polynomials over GF(2). 1 Introduction Four of the most natural group actions on strings over a fixed alphabet are: (a) leaving the string unchanged, (b) reversing the string, (c) rotat...

A non-empty word w of {a, b}* is a Lyndon word if and only if it is strictly smaller for the lexicographical order than any of its proper suffixes. Such a word w is either a letter or admits a standard factorization uv where v is its smallest proper suffix. For any Lyndon word v, we show that the set of Lyndon words having v as right factor of the standard factorization is rational and compute explicitly the associated generating function. Next we establish that, for the uniform distribution over the Lyndon words of length n, the average length of the right factor v of the standard factorization is asymptotically 3n/4.