Discordant permutations are two permutations of the same set that differ at every position. For permutations
of 
,
write 
and 
.
Here 
and 
denote the entries in position 
of the respective permutations,
so the condition is
 |
for every 
(Wahid 2001, p. 57). For example, the permutations 
and 
are discordant, although neither is a derangement.
A permutation discordant with the identity permutation is a derangement. More generally,
and 
are discordant iff 
is a derangement.
Consequently, the number of permutations discordant
with any fixed permutation of 
elements is
 |
where 
is the subfactorial and 
is the factorial (Wahid 2001,
pp. 57-58). More generally, the partial derangement
counts enumerate permutations agreeing with a fixed
reference permutation in exactly 
positions.
For 
,
the reduced ménage number 
of the married couples
problem counts permutations discordant with both
the identity permutation and the cyclic
permutation 
modulo 
.
Thus the two forbidden positions are expressed by 
and 
modulo 
(Wahid 2001, p. 58).
." Revista de Matemática: Teoría y Aplicaciones 8,
47-62, 2001.