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, ....
For 
,
these agree with the even-permutation counts
(OEIS A001710), while for 
there is one even permutation
and no odd permutation.
