Page 76 - A Course in Linear Algebra with Applications
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60 Chapter Three: Determinants
which is written
n!
and referred to as "n factorial". Thus we can state the follow-
ing basic result.
Theorem 3.1.1
The number of permutations of the integers 1,2,... ,n equals
l
n! = n ( n - )---2- 1.
Even and odd permutations
A permutation of the integers 1,2, ...,n is called even
or odd according to whether the number of inversions of the
natural order 1,2,... ,n that are present in the permutation
is even or odd respectively. For example, the permutation 1,
3, 2 involves a single inversion, for 3 comes before 2; so this
is an odd permutation. For permutations of longer sequences
of integers it is advantageous to count inversions by means of
what is called a crossover diagram. This is best explained by
an example.
Example 3.1.1
Is the permutation 8, 3, 2, 6, 5, 1, 4, 7 even or odd?
The procedure is to write the integers 1 through 8 in the
natural order in a horizontal line, and then to write down the
entries of the permutation in the line below. Join each integer
i in the top line to the same integer i where it appears in the
bottom line, taking care to avoid multiple intersections. The
number of intersections or crossovers will be the number of
inversions present in the permutation:
