Unit IV: Determinants
1. Permutations
Definition: A
permutation
of a set
S
is a function from
S
to
S
that is onetoone and
onto. If
S
has
n
elements, then there are
n
! permutations of
S
.
The notation
π
:
{
1
, . . . , n
} 7→ {
k
1
. . . k
n
}
refers to the permutation
π
of the integers
1
, . . . , n
for which
π
(1) =
k
1
,
π
(2) =
k
2
, and so on, to
π
(
n
) =
k
n
.
Example: There are two permutations of the set
{
1
,
2
}
, namely the identity
{
1
,
2
} 7→
{
1
,
2
}
and the switch
{
1
,
2
} 7→ {
2
,
1
}
.
There are six permutations of the set
{
1
,
2
,
3
}
.
They are the identity
{
1
,
2
,
3
} 7→ {
1
,
2
,
3
}
, two circular shifts
{
1
,
2
,
3
} 7→ {
2
,
3
,
1
}
and
{
1
,
2
,
3
} 7→ {
3
,
1
,
2
}
, and three switches
{
1
,
2
,
3
} 7→ {
2
,
1
,
3
}
,
{
1
,
2
,
3
} 7→ {
3
,
2
,
1
}
, and
{
1
,
2
,
3
} 7→ {
1
,
3
,
2
}
.
The composition
σ
◦
π
of two permutations
σ
and
π
is a permutation. Each permutation
π
has an inverse permutation
π

1
, defined so that
π

1
◦
π
=
π
◦
π

1
= the identity permutation.
We will refer to a permutation that simply switches two values as a
switch
. (The math
ematical word is
transposition
.)
Any permutation is a composition of switches.
For in
stance, the circular shift
{
1
,
2
,
3
} 7→ {
2
,
3
,
1
}
is a composition of two switches, the switch
{
1
,
2
,
3
} 7→ {
2
,
1
,
3
}
of the first two values, followed by the switch
{
1
,
2
,
3
} 7→ {
3
,
2
,
1
}
of the
first and third values.
The
inversion number
of a permutation
π
:
{
1
, . . . , n
} 7→ {
π
(1)
, . . . π
(
n
)
}
is the number
of pairs (
j, k
) such that
j < k
but
π
(
j
)
> π
(
k
).
Example: The identity permutation of
{
1
,
2
,
3
}
has inversion number 0, the circular shifts
have inversion number 2, and the switches have inversion numbers 1 or 3. For instance, the
circular shift
{
1
,
2
,
3
} 7→ {
2
,
3
,
1
}
has inversion number 2, corresponding to the inversions
π
(1) = 2
>
1 =
π
(3) and
π
(2) = 3
>
1 =
π
(3). The switch
{
1
,
2
,
3
} 7→ {
2
,
1
,
3
}
has inversion
number 1, corresponding to the inversion
π
(1) = 2
>
1 =
π
(2).
Definition: An
even permutation
is a permutation whose inversion number is even, and
an
odd permutation
is a permutation whose inversion number is odd. The
parity of a per
mutation
is even or odd according to whether its inversion number is even or odd.
Definition: The
sign of a permutation
is +1 if the permutation is even and

1 if the
permutation is odd. We denote the sign of a permutation
π
by sgn(
π
).
Theorem 1.
If we switch two values of a permutation, the sign changes.
Proofsketch.
If we switch the values of adjacent integers, say the values
π
(
j
) and
π
(
j
+1),
the inversion number of the new permutation changes by +1 or

1, depending on whether
π
(
j
)
< π
(
j
+ 1) or
π
(
j
)
> π
(
j
+ 1). Switching two values of arbitrary integers, say
π
(
j
) and
π
(
k
) can be done through a succession of an odd number of switches of values of adjacent
integers.
Procedure.
The theorem provides an easy way to determine the parity of a permutation.
Count the number of switches you need to get from the identity permutation to
π
. If you
can do it with an even number of switches, then
π
is even. If you can do it with an odd
number of switches, then
π
is odd.