Summary: 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.

05A10, 20B35

inversions, permutations, symmetric groups, alternating groups, dihedral groups