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Permutation statistics on the alternating group. (English) Zbl 1057.05004
Let S\(_n\) and A\(_n\) be the symmetric and the alternating group on a set of cardinality \(n\). One is interested in the refined count of permutations according to nonnegative, integer-valued combinatorial parameters. For example, the number of inversions in a permutation \(\pi\) – namely its length – is such a parameter (this is the minimal number of adjacent transpositions, which generate the subgroup containing \(\pi\)). Another important parameter is the so-called major index of \(\pi\), introduced by MacMahon in his ‘Combinatorial analysis’. Two parameters that have the same generating function are said to be equidistributed. MacMahon proved that the length and the major-index statistics are equidistributed on S\(_n\). The above statistics fail to be equidistributed on A\(_n\). The main goal of this paper is to find statistics on A\(_n\) which are natural analogues of the above S\(_n\) statistics and are equidistributed on A\(_n\), yielding analogous identities for their generating functions. To obtain additional information, we suggest the reader to read this interesting paper.

MSC:
05A05 Permutations, words, matrices
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