A bijection between words and multisets of necklaces

Abstract

Two of the present authors have given in 1993 a bijection $\Phi$ between words on a totally ordered alphabet and multisets of primitive necklaces. At the same time and independently, Burrows and Wheeler gave a data compression algorithm which turns out to be a particular case of the inverse of $\Phi$. In the present article, we show that if one replaces in $\Phi$ the standard permutation of a word by the co-standard one (reading the word from right to left), then the inverse bijection is computed using the alternate lexicographic order (which is the order of real numbers given by continued fractions) on necklaces, instead of the lexicographic order as for ${\Phi }^{-1}$. The image of the new bijection, instead of being as for $\Phi$ the set of all multisets of primitive necklaces, is a special set of multisets of necklaces (not all primitive); it turns out that this set is naturally linked to the decomposition of the enveloping algebra of the oddly generated free Lie superalgebra, induced by the Poincaré–Birkhoff–Witt theorem.