A bijection between words and multisets of necklaces

Abstract

In [4] has been proved the bijectivity of a natural mapping Φ from words on a totally ordered alphabet onto multisets of primitive necklaces (circular words) on this alphabet. This mapping has many enumerative applications; among them, the fact that the number of permutations in a given conjugation class and with a given descent set is equal to the scalar product of two representations naturally associated to the class and the set. The direct mapping is defined by associating to each word its standard permutation, and then replacing in the cycles of the latter each element by the corresponding letter of the word. The inverse mapping φ−1 is constructed using the lexicographic order of infinite words. In the present article, we replace the standard permutation by the costandard permutation. That is, the permutation obtained by numbering the positions in the word from right to left (instead from left to right as it is done for the standard permutation). This a priori useless generalization has however striking properties. Indeed, it induces a bijection Ξ between words and multisets of necklaces, which are intimately related to the PoincaréBirkhoff-Witt theorem applied to the free Lie superalgebra (instead of the free Lie algebra as it is the case for Φ). See Section 3 for the exact description of these multisets. Another striking property of the bijection Ξ is hat the inverse bijection uses, instead of the lexicographic order, the alternate lexicographic order of infinite words; this means that one compares the first letters of the infinite words for the given order of the alphabet, then if equality, the second letters for the opposite order, and so on. The alternate lexicographic order is very natural. Indeed it corresponds to the order of real numbers given by their expansion into continued fractions. This is well-known and used implicitly very often, see for example the book [3] on the the Markoff and Lagrange spectra. The proof of the bijectivity of Ξ that we give has as a byproduct a new proof of the bijectivity of Φ, that we give in Section 2. In Section 5 we recall the symmetric functions that are induced by Φ, and then show that the symmetric functions induced by Ξ are related to the free Lie superalgebra and equal to the image of the formers under the fundamental involution of symmetric functions.

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