Séminaire Lotharingien de Combinatoire, B22a (1989), 9
[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 414/S-22, p.
Étude modulo n des statistiques mahoniennes
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
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.
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