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.
