Inversions of Permutations in Symmetric, Alternating, and Dihedral Groups

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 Sn 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. 1