Using the combinatorial techniques developed by Barcelo and Bergeron, we construct a Bn-module, called (n), related to the Orlik-Solomon algebra of the hyperoctahedral hyperplane complements lattice (os(Bn)) [10, 13]. The Bn-modules (n) and os(Bn) are analogous to the modules for the symmetric group which occur in the context of the free Lie algebra and the partition lattice. In particular we show that the module (n) is the transpose of the module os(Bn) tensored by a sign representation. As a by-product we show that the action of Bn on a natural basis of (n) is block triangular. The blocks are indexed by the conjugacy classes of Bn and have dimension equal to the number of elements in such a class. We also compute the characters of this action restricted to each block.
The algebraic structure underlying non-commutative Lie–Butcher series is the free Lie algebra over ordered trees. In this paper we present a characterization of this algebra in terms of balanced Lyndon words over a binary alphabet. This yields a systematic manner of enumerating terms in non-commutative Lie–Butcher series.
An O(logn log log n) CRCW PRAM algorithm using O(n/log n) processors for computing the unique Lyndon factorization of a word of length n over an unbounded alphabet is presented; this improves the bounds given by Apostolico and Crochemore (1989). Moreover, in the case of fixed alphabets the CRCW PRAM algorithm is optimal (linear cost), requiring O(log n) units of time.
Two different factorizations of the Fibonacci infinite word were given independently in Wen and Wen (1994) and Melançon (1996). In a certain sense, these factorizations reveal a selfsimilarity property of the Fibonacci word. We first describe the intimate links between these two factorizations. We then propose a generalization to characteristic sturmian words.
Résumé
Deux factorisations du mot de Fibonacci ont été données dans deux articles indépendants, Wen and Wen (1994) and Melançon (1996). Ces factorisations décrivent, d'une certaine manière, une propriété d'auto-similarité du mot de Fibonacci. Nous décrivons d'abord les liens étroits entre ces deux factorisations. Puis nous proposons une généralisation aux mots sturmiens caractéristiques.
Let G = (X,E) be a finite connected (undirected) graph; a permutation σ over X is said to be compatible with G when every vertex x different from σ(x) is adjacent to σ(x); l(G) denotes the minimum k such that every permutation over X can be decomposed into a product of k permutations compatible with G. It is shown that, among all trees with X as set of vertices, the chain has the smallest l(G).
Christophe Hohlweg, a, b and Christophe Reutenauer, , a, b, 1
a IRMA, 7 rue René Descartes, 67084, Strasbourg, France b Université du Québec à Montréal, Département de Mathématiques, case postale 8888, Succursale A Centre-ville, Montréal, Qué., Canada H3C 3P8
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