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Inversions of Permutations in Symmetric, Alternating, and Dihedral Groups
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Dexter Jane L. Indong and Gilbert R. Peralta

Department of Mathematics and Computer Science

University of the Philippines Baguio

Governor Pack Road

Baguio City 2600

Philippines

**Abstract:**

We use two methods to obtain a formula relating the total number of
inversions of all permutations and the corresponding order of
symmetric, alternating, and dihedral groups. First, we define an
equivalence relation on the symmetric group **S**_{n} and
consider each element in each equivalence class as a permutation of a
proper subset of {1,2, ... , *n*}. Second, we look at certain
properties of a backward permutation, a permutation obtained by
reversing the row images of a given permutation. Lastly, we employ
the first method to obtain a recursive formula corresponding to the
number of permutations with *k* inversions.

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(Concerned with sequences
A001809 and
A006002.)

Received May 11 2008;
revised version received September 29 2008.
Published in *Journal of Integer Sequences*, October 4 2008.

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