Modulo-n study of Mahonian statistics

Jacques Désarmémien

Séminaire lotharingien de combinatoire, Tome 22 (1989)

Résumé

Under the generic name of mahonian statistics we understand a certain class of order statistics defined on sets of words of length n, such as the number of inversions, the major index and also the inverse major index for permutations. We can also impose some constraints on the up-down sequences of the permutations and refer to a celebrated result by Foata and Schützenberger that establishes the equidistribution of the inversion number and the inverse major index for each class of permutations with a given up-down sequence. This allows us to transfer the study of those statistics on statistics on Young tableaux and then use Classical Algebra techniques.

The purpose of this paper is to prove an equidistribution property for Young tableaux of a given form with respect to their major indices modulo n. The same property holds for all permutation statistics. Several other analogous results are also derived.

We also derive the explict decomposition of the representation of the symmetric group on the free Lie algebra associated with the partition n. The latter result due to Kraskiewicz and Weyman is quoted by Reutenauer.

We make use of the link of certain characters of the symmetric group with the major indices of Young tableaux, and also of a lemma of arithmetic nature.