Journal of Integer Sequences, Vol. 11 (2008), Article 08.4.3

Inversions of Permutations in Symmetric, Alternating, and Dihedral Groups

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


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.

Full version:  pdf,    dvi,    ps,    latex    

(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.

Return to Journal of Integer Sequences home page