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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
##### Keywords:
major index; equidistributed; statistics
Full Text:
##### References:
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