Geodesic
Inversions of permutations in symmetric, alternating, and dihedral groups
Journal of integer sequences, Tome 11 (2008) no. 4
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 ${1,2, \dots , n}$. 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.
Classification : 05A10, 20B35
Keywords: inversions, permutations, symmetric groups, alternating groups, dihedral groups 
@article{JIS_2008__11_4_a1,
     author = {Indong,  Dexter Jane L. and Peralta,  Gilbert R.},
     title = {Inversions of permutations in symmetric, alternating, and dihedral groups},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {4},
     zbl = {1148.05004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_4_a1/}
}
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Indong,  Dexter Jane L.; Peralta,  Gilbert R. Inversions of permutations in symmetric, alternating, and dihedral groups. Journal of integer sequences, Tome 11 (2008) no. 4. http://geodesic.mathdoc.fr/item/JIS_2008__11_4_a1/