An odd permutation is a permutation obtainable from an odd number of two-element swaps, i.e., a permutation
with permutation symbol equal to 
. For initial set 
1,2,3,4
, the twelve odd permutations are those with one swap (
1,2,4,3
, 
1,3,2,4
, 
1,4,3,2
, 
2,1,3,4
, 
3,2,1,4
, 
4,2,3,1
) and those with three swaps (
2,3,4,1
, 
2,4,1,3
, 
3,1,4,2
, 
3,4,2,1
, 
4,1,2,3
, 
4,3,1,2
).
For a set of 
elements and 
,
there are 
odd permutations (D'Angelo and West 2000, p. 111), which is the same as the
number of even permutations. For 
, 2, ..., the numbers are given by 0, 1, 3, 12, 60, 360,
2520, 20160, 181440, ... (OEIS A001710).
